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Theorem noinfep 4612
Description: Using the Axiom of Regularity in the form zfregfr 4573, show that there are no infinite descending e. -chains. Proposition 7.34 of [TakeutiZaring] p. 44.
Assertion
Ref Expression
noinfep |- (F Fn om -> E.x e. om -. (F` suc x) e. (F` x))
Distinct variable group:   x,F
Proof of Theorem noinfep
StepHypRef Expression
1 zfregfr 4573 . . . 4 |- E Fr ran F
2 ssid 2070 . . . . 5 |- ran F (_ ran F
3 fri 2908 . . . . 5 |- (((ran F e. V /\ E Fr ran F) /\ (ran F (_ ran F /\ ran F =/= (/))) -> E.y e. ran FA.z e. ran F -. zEy)
42, 3mpanr1 707 . . . 4 |- (((ran F e. V /\ E Fr ran F) /\ ran F =/= (/)) -> E.y e. ran FA.z e. ran F -. zEy)
51, 4mpanl2 705 . . 3 |- ((ran F e. V /\ ran F =/= (/)) -> E.y e. ran FA.z e. ran F -. zEy)
6 funrnex 3599 . . . 4 |- (dom F e. V -> (Fun F -> ran F e. V))
7 fndm 3573 . . . . 5 |- (F Fn om -> dom F = om)
8 omex 4599 . . . . 5 |- om e. V
97, 8syl6eqel 1548 . . . 4 |- (F Fn om -> dom F e. V)
10 fnfun 3571 . . . 4 |- (F Fn om -> Fun F)
116, 9, 10sylc 68 . . 3 |- (F Fn om -> ran F e. V)
12 peano1 3139 . . . . . . 7 |- (/) e. om
13 eleq2 1527 . . . . . . 7 |- (dom F = om -> ((/) e. dom F <-> (/) e. om))
1412, 13mpbiri 194 . . . . . 6 |- (dom F = om -> (/) e. dom F)
15 ne0i 2276 . . . . . 6 |- ((/) e. dom F -> dom F =/= (/))
1614, 15syl 10 . . . . 5 |- (dom F = om -> dom F =/= (/))
17 dm0rn0 3319 . . . . . 6 |- (dom F = (/) <-> ran F = (/))
1817necon3bii 1590 . . . . 5 |- (dom F =/= (/) <-> ran F =/= (/))
1916, 18sylib 198 . . . 4 |- (dom F = om -> ran F =/= (/))
207, 19syl 10 . . 3 |- (F Fn om -> ran F =/= (/))
215, 11, 20sylanc 471 . 2 |- (F Fn om -> E.y e. ran FA.z e. ran F -. zEy)
22 fvelrnb 3745 . . . . . . 7 |- (F Fn om -> (y e. ran F <-> E.x e. om (F` x) = y))
2322adantr 389 . . . . . 6 |- ((F Fn om /\ A.z e. ran F -. zEy) -> (y e. ran F <-> E.x e. om (F` x) = y))
24 eleq2 1527 . . . . . . . . . . . 12 |- ((F` x) = y -> ((F` suc x) e. (F` x) <-> (F` suc x) e. y))
2524negbid 609 . . . . . . . . . . 11 |- ((F` x) = y -> (-. (F` suc x) e. (F` x) <-> -. (F` suc x) e. y))
26 eleq1 1526 . . . . . . . . . . . . . 14 |- (z = (F` suc x) -> (z e. y <-> (F` suc x) e. y))
27 epel 2823 . . . . . . . . . . . . . 14 |- (zEy <-> z e. y)
2826, 27syl5bb 530 . . . . . . . . . . . . 13 |- (z = (F` suc x) -> (zEy <-> (F` suc x) e. y))
2928negbid 609 . . . . . . . . . . . 12 |- (z = (F` suc x) -> (-. zEy <-> -. (F` suc x) e. y))
3029rcla4va 1866 . . . . . . . . . . 11 |- (((F` suc x) e. ran F /\ A.z e. ran F -. zEy) -> -. (F` suc x) e. y)
3125, 30syl5bir 210 . . . . . . . . . 10 |- ((F` x) = y -> (((F` suc x) e. ran F /\ A.z e. ran F -. zEy) -> -. (F` suc x) e. (F` x)))
32 fnfvelrn 3798 . . . . . . . . . . . 12 |- ((F Fn om /\ suc x e. om) -> (F` suc x) e. ran F)
3332adantlr 393 . . . . . . . . . . 11 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> (F` suc x) e. ran F)
34 simplr 413 . . . . . . . . . . 11 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> A.z e. ran F -. zEy)
3533, 34jca 288 . . . . . . . . . 10 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> ((F` suc x) e. ran F /\ A.z e. ran F -. zEy))
3631, 35syl5 21 . . . . . . . . 9 |- ((F` x) = y -> (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> -. (F` suc x) e. (F` x)))
37 peano2 3140 . . . . . . . . 9 |- (x e. om -> suc x e. om)
3836, 37sylan2i 465 . . . . . . . 8 |- ((F` x) = y -> (((F Fn om /\ A.z e. ran F -. zEy) /\ x e. om) -> -. (F` suc x) e. (F` x)))
3938com12 11 . . . . . . 7 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ x e. om) -> ((F` x) = y -> -. (F` suc x) e. (F` x)))
4039r19.22dva 1731 . . . . . 6 |- ((F Fn om /\ A.z e. ran F -. zEy) -> (E.x e. om (F` x) = y -> E.x e. om -. (F` suc x) e. (F` x)))
4123, 40sylbid 203 . . . . 5 |- ((F Fn om /\ A.z e. ran F -. zEy) -> (y e. ran F -> E.x e. om -. (F` suc x) e. (F` x)))
4241ex 373 . . . 4 |- (F Fn om -> (A.z e. ran F -. zEy -> (y e. ran F -> E.x e. om -. (F` suc x) e. (F` x))))
4342com23 32 . . 3 |- (F Fn om -> (y e. ran F -> (A.z e. ran F -. zEy -> E.x e. om -. (F` suc x) e. (F` x))))
4443r19.23adv 1738 . 2 |- (F Fn om -> (E.y e. ran FA.z e. ran F -. zEy -> E.x e. om -. (F` suc x) e. (F` x)))
4521, 44mpd 26 1 |- (F Fn om -> E.x e. om -. (F` suc x) e. (F` x))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  E.wrex 1638  Vcvv 1802   (_ wss 2037  (/)c0 2270   class class class wbr 2609  Ecep 2819   Fr wfr 2905  suc csuc 2940  omcom 3121  dom cdm 3160  ran crn 3161  Fun wfun 3166   Fn wfn 3167  ` cfv 3172
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188
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