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Theorem tz7.44-1 3913
Description: The value of F at (/). Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49.
Hypotheses
Ref Expression
tz7.44.1 |- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
tz7.44.2 |- F Fn On
tz7.44.3 |- (x e. On -> (F` x) = (G` (F |` x)))
tz7.44.4 |- A e. V
Assertion
Ref Expression
tz7.44-1 |- (F` (/)) = A
Distinct variable groups:   x,y,A   x,F   x,G   y,H
Proof of Theorem tz7.44-1
StepHypRef Expression
1 0elon 3012 . . 3 |- (/) e. On
2 fveq2 3709 . . . . 5 |- (x = (/) -> (F` x) = (F` (/)))
3 reseq2 3353 . . . . . 6 |- (x = (/) -> (F |` x) = (F |` (/)))
43fveq2d 3713 . . . . 5 |- (x = (/) -> (G` (F |` x)) = (G` (F |` (/))))
52, 4eqeq12d 1481 . . . 4 |- (x = (/) -> ((F` x) = (G` (F |` x)) <-> (F` (/)) = (G` (F |` (/)))))
6 tz7.44.3 . . . 4 |- (x e. On -> (F` x) = (G` (F |` x)))
75, 6vtoclga 1843 . . 3 |- ((/) e. On -> (F` (/)) = (G` (F |` (/))))
81, 7ax-mp 7 . 2 |- (F` (/)) = (G` (F |` (/)))
9 res0 3355 . . 3 |- (F |` (/)) = (/)
109fveq2i 3712 . 2 |- (G` (F |` (/))) = (G` (/))
11 tz7.44.1 . . . 4 |- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
1211tz7.44lem1 3912 . . 3 |- Fun G
13 3mix1 813 . . . . . 6 |- ((x = (/) /\ y = A) -> ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x)))
1413ssopab2i 2812 . . . . 5 |- {<.x, y>. | (x = (/) /\ y = A)} (_ {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
1514, 11sseqtr4 2084 . . . 4 |- {<.x, y>. | (x = (/) /\ y = A)} (_ G
16 0ex 2701 . . . . . 6 |- (/) e. V
17 tz7.44.4 . . . . . 6 |- A e. V
18 eqeq1 1473 . . . . . . 7 |- (x = (/) -> (x = (/) <-> (/) = (/)))
1918anbi1d 615 . . . . . 6 |- (x = (/) -> ((x = (/) /\ y = A) <-> ((/) = (/) /\ y = A)))
20 eqeq1 1473 . . . . . . 7 |- (y = A -> (y = A <-> A = A))
2120anbi2d 614 . . . . . 6 |- (y = A -> (((/) = (/) /\ y = A) <-> ((/) = (/) /\ A = A)))
2216, 17, 19, 21opelopab 2809 . . . . 5 |- (<.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)} <-> ((/) = (/) /\ A = A))
23 eqid 1468 . . . . 5 |- (/) = (/)
24 eqid 1468 . . . . 5 |- A = A
2522, 23, 24mpbir2an 728 . . . 4 |- <.(/), A>. e. {<.x, y>. | (x = (/) /\ y = A)}
2615, 25sselii 2056 . . 3 |- <.(/), A>. e. G
2717funopfv 3736 . . 3 |- (Fun G -> (<.(/), A>. e. G -> (G` (/)) = A))
2812, 26, 27mp2 43 . 2 |- (G` (/)) = A
298, 10, 283eqtr 1491 1 |- (F` (/)) = A
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270  <.cop 2401  U.cuni 2493  {copab 2656  Oncon0 2938  Lim wlim 2939  dom cdm 3160  ran crn 3161   |` cres 3162  Fun wfun 3166   Fn wfn 3167  ` cfv 3172
This theorem is referenced by:  rdg0 3926
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-9 962  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-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857
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-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  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-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-fv 3188
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