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Theorem alephfplem4 8505
Description: Lemma for alephfp 8506. (Contributed by NM, 5-Nov-2004.)
Hypothesis
Ref Expression
alephfplem.1  |-  H  =  ( rec ( aleph ,  om )  |`  om )
Assertion
Ref Expression
alephfplem4  |-  U. ( H " om )  e. 
ran  aleph

Proof of Theorem alephfplem4
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 frfnom 7118 . . . . 5  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
2 alephfplem.1 . . . . . 6  |-  H  =  ( rec ( aleph ,  om )  |`  om )
32fneq1i 5681 . . . . 5  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
41, 3mpbir 209 . . . 4  |-  H  Fn  om
52alephfplem3 8504 . . . . 5  |-  ( z  e.  om  ->  ( H `  z )  e.  ran  aleph )
65rgen 2817 . . . 4  |-  A. z  e.  om  ( H `  z )  e.  ran  aleph
7 ffnfv 6058 . . . 4  |-  ( H : om --> ran  aleph  <->  ( H  Fn  om  /\  A. z  e.  om  ( H `  z )  e.  ran  aleph
) )
84, 6, 7mpbir2an 920 . . 3  |-  H : om
--> ran  aleph
9 ssun2 3664 . . 3  |-  ran  aleph  C_  ( om  u.  ran  aleph )
10 fss 5745 . . 3  |-  ( ( H : om --> ran  aleph  /\  ran  aleph  C_  ( om  u.  ran  aleph
) )  ->  H : om --> ( om  u.  ran  aleph ) )
118, 9, 10mp2an 672 . 2  |-  H : om
--> ( om  u.  ran  aleph
)
12 peano1 6718 . . 3  |-  (/)  e.  om
132alephfplem1 8502 . . 3  |-  ( H `
 (/) )  e.  ran  aleph
14 fveq2 5872 . . . . 5  |-  ( z  =  (/)  ->  ( H `
 z )  =  ( H `  (/) ) )
1514eleq1d 2526 . . . 4  |-  ( z  =  (/)  ->  ( ( H `  z )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
1615rspcev 3210 . . 3  |-  ( (
(/)  e.  om  /\  ( H `  (/) )  e. 
ran  aleph )  ->  E. z  e.  om  ( H `  z )  e.  ran  aleph
)
1712, 13, 16mp2an 672 . 2  |-  E. z  e.  om  ( H `  z )  e.  ran  aleph
18 omex 8077 . . 3  |-  om  e.  _V
19 cardinfima 8495 . . 3  |-  ( om  e.  _V  ->  (
( H : om --> ( om  u.  ran  aleph )  /\  E. z  e.  om  ( H `  z )  e.  ran  aleph )  ->  U. ( H " om )  e. 
ran  aleph ) )
2018, 19ax-mp 5 . 2  |-  ( ( H : om --> ( om  u.  ran  aleph )  /\  E. z  e.  om  ( H `  z )  e.  ran  aleph )  ->  U. ( H " om )  e. 
ran  aleph )
2111, 17, 20mp2an 672 1  |-  U. ( H " om )  e. 
ran  aleph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    u. cun 3469    C_ wss 3471   (/)c0 3793   U.cuni 4251   ran crn 5009    |` cres 5010   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594   omcom 6699   reccrdg 7093   alephcale 8334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338
This theorem is referenced by:  alephfp  8506  alephfp2  8507
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