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Theorem alephfplem4 7734
Description: Lemma for alephfp 7735. (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 6447 . . . . 5  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
2 alephfplem.1 . . . . . 6  |-  H  =  ( rec ( aleph ,  om )  |`  om )
32fneq1i 5338 . . . . 5  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
41, 3mpbir 200 . . . 4  |-  H  Fn  om
52alephfplem3 7733 . . . . 5  |-  ( z  e.  om  ->  ( H `  z )  e.  ran  aleph )
65rgen 2608 . . . 4  |-  A. z  e.  om  ( H `  z )  e.  ran  aleph
7 ffnfv 5685 . . . 4  |-  ( H : om --> ran  aleph  <->  ( H  Fn  om  /\  A. z  e.  om  ( H `  z )  e.  ran  aleph
) )
84, 6, 7mpbir2an 886 . . 3  |-  H : om
--> ran  aleph
9 ssun2 3339 . . 3  |-  ran  aleph  C_  ( om  u.  ran  aleph )
10 fss 5397 . . 3  |-  ( ( H : om --> ran  aleph  /\  ran  aleph  C_  ( om  u.  ran  aleph
) )  ->  H : om --> ( om  u.  ran  aleph ) )
118, 9, 10mp2an 653 . 2  |-  H : om
--> ( om  u.  ran  aleph
)
12 peano1 4675 . . 3  |-  (/)  e.  om
132alephfplem1 7731 . . 3  |-  ( H `
 (/) )  e.  ran  aleph
14 fveq2 5525 . . . . 5  |-  ( z  =  (/)  ->  ( H `
 z )  =  ( H `  (/) ) )
1514eleq1d 2349 . . . 4  |-  ( z  =  (/)  ->  ( ( H `  z )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
1615rspcev 2884 . . 3  |-  ( (
(/)  e.  om  /\  ( H `  (/) )  e. 
ran  aleph )  ->  E. z  e.  om  ( H `  z )  e.  ran  aleph
)
1712, 13, 16mp2an 653 . 2  |-  E. z  e.  om  ( H `  z )  e.  ran  aleph
18 omex 7344 . . 3  |-  om  e.  _V
19 cardinfima 7724 . . 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 653 1  |-  U. ( H " om )  e. 
ran  aleph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    C_ wss 3152   (/)c0 3455   U.cuni 3827   omcom 4656   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255   reccrdg 6422   alephcale 7569
This theorem is referenced by:  alephfp  7735  alephfp2  7736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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