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Theorem alephfplem4 7980
Description: Lemma for alephfp 7981. (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 6684 . . . . 5  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
2 alephfplem.1 . . . . . 6  |-  H  =  ( rec ( aleph ,  om )  |`  om )
32fneq1i 5531 . . . . 5  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
41, 3mpbir 201 . . . 4  |-  H  Fn  om
52alephfplem3 7979 . . . . 5  |-  ( z  e.  om  ->  ( H `  z )  e.  ran  aleph )
65rgen 2763 . . . 4  |-  A. z  e.  om  ( H `  z )  e.  ran  aleph
7 ffnfv 5886 . . . 4  |-  ( H : om --> ran  aleph  <->  ( H  Fn  om  /\  A. z  e.  om  ( H `  z )  e.  ran  aleph
) )
84, 6, 7mpbir2an 887 . . 3  |-  H : om
--> ran  aleph
9 ssun2 3503 . . 3  |-  ran  aleph  C_  ( om  u.  ran  aleph )
10 fss 5591 . . 3  |-  ( ( H : om --> ran  aleph  /\  ran  aleph  C_  ( om  u.  ran  aleph
) )  ->  H : om --> ( om  u.  ran  aleph ) )
118, 9, 10mp2an 654 . 2  |-  H : om
--> ( om  u.  ran  aleph
)
12 peano1 4856 . . 3  |-  (/)  e.  om
132alephfplem1 7977 . . 3  |-  ( H `
 (/) )  e.  ran  aleph
14 fveq2 5720 . . . . 5  |-  ( z  =  (/)  ->  ( H `
 z )  =  ( H `  (/) ) )
1514eleq1d 2501 . . . 4  |-  ( z  =  (/)  ->  ( ( H `  z )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
1615rspcev 3044 . . 3  |-  ( (
(/)  e.  om  /\  ( H `  (/) )  e. 
ran  aleph )  ->  E. z  e.  om  ( H `  z )  e.  ran  aleph
)
1712, 13, 16mp2an 654 . 2  |-  E. z  e.  om  ( H `  z )  e.  ran  aleph
18 omex 7590 . . 3  |-  om  e.  _V
19 cardinfima 7970 . . 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 8 . 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 654 1  |-  U. ( H " om )  e. 
ran  aleph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    u. cun 3310    C_ wss 3312   (/)c0 3620   U.cuni 4007   omcom 4837   ran crn 4871    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446   reccrdg 6659   alephcale 7815
This theorem is referenced by:  alephfp  7981  alephfp2  7982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-card 7818  df-aleph 7819
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