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Theorem alephfplem3 4870
Description: Lemma for alephfp 4872.
Hypothesis
Ref Expression
alephfplem.1 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
Assertion
Ref Expression
alephfplem3 |- (v e. om -> (H` v) e. ran aleph)
Distinct variable groups:   x,y,v   v,H
Proof of Theorem alephfplem3
StepHypRef Expression
1 equid 1122 . 2 |- y = y
2 fveq2 3709 . . . 4 |- (v = (/) -> (H` v) = (H` (/)))
32eleq1d 1532 . . 3 |- (v = (/) -> ((H` v) e. ran aleph <-> (H` (/)) e. ran aleph))
4 fveq2 3709 . . . 4 |- (v = w -> (H` v) = (H` w))
54eleq1d 1532 . . 3 |- (v = w -> ((H` v) e. ran aleph <-> (H` w) e. ran aleph))
6 fveq2 3709 . . . 4 |- (v = suc w -> (H` v) = (H` suc w))
76eleq1d 1532 . . 3 |- (v = suc w -> ((H` v) e. ran aleph <-> (H` suc w) e. ran aleph))
8 alephfplem.1 . . . . 5 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
98alephfplem1 4868 . . . 4 |- (H` (/)) e. ran aleph
109a1i 8 . . 3 |- (y = y -> (H` (/)) e. ran aleph)
118alephfplem2 4869 . . . . . 6 |- (w e. om -> (H` suc w) = (aleph` (H` w)))
1211eleq1d 1532 . . . . 5 |- (w e. om -> ((H` suc w) e. ran aleph <-> (aleph` (H` w)) e. ran aleph))
13 alephsson 4866 . . . . . . 7 |- ran aleph (_ On
1413sseli 2055 . . . . . 6 |- ((H` w) e. ran aleph -> (H` w) e. On)
15 alephfnon 4834 . . . . . . 7 |- aleph Fn On
16 fnfvelrn 3798 . . . . . . 7 |- ((aleph Fn On /\ (H` w) e. On) -> (aleph` (H` w)) e. ran aleph)
1715, 16mpan 693 . . . . . 6 |- ((H` w) e. On -> (aleph` (H` w)) e. ran aleph)
1814, 17syl 10 . . . . 5 |- ((H` w) e. ran aleph -> (aleph` (H` w)) e. ran aleph)
1912, 18syl5bir 210 . . . 4 |- (w e. om -> ((H` w) e. ran aleph -> (H` suc w) e. ran aleph))
2019a1d 12 . . 3 |- (w e. om -> (y = y -> ((H` w) e. ran aleph -> (H` suc w) e. ran aleph)))
213, 5, 7, 10, 20finds2 3148 . 2 |- (v e. om -> (y = y -> (H` v) e. ran aleph))
221, 21mpi 44 1 |- (v e. om -> (H` v) e. ran aleph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  (/)c0 2270  {copab 2656  Oncon0 2938  suc csuc 2940  omcom 3121  ran crn 3161   |` cres 3162   Fn wfn 3167  ` cfv 3172  reccrdg 3916  alephcale 4786
This theorem is referenced by:  alephfplem4 4871  alephfp 4872
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-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716
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-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  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-int 2524  df-iun 2558  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789
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