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Theorem alephfplem4 7923
Description: Lemma for alephfp 7924. (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 6630 . . . . 5  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
2 alephfplem.1 . . . . . 6  |-  H  =  ( rec ( aleph ,  om )  |`  om )
32fneq1i 5481 . . . . 5  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
41, 3mpbir 201 . . . 4  |-  H  Fn  om
52alephfplem3 7922 . . . . 5  |-  ( z  e.  om  ->  ( H `  z )  e.  ran  aleph )
65rgen 2716 . . . 4  |-  A. z  e.  om  ( H `  z )  e.  ran  aleph
7 ffnfv 5835 . . . 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 3456 . . 3  |-  ran  aleph  C_  ( om  u.  ran  aleph )
10 fss 5541 . . 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 4806 . . 3  |-  (/)  e.  om
132alephfplem1 7920 . . 3  |-  ( H `
 (/) )  e.  ran  aleph
14 fveq2 5670 . . . . 5  |-  ( z  =  (/)  ->  ( H `
 z )  =  ( H `  (/) ) )
1514eleq1d 2455 . . . 4  |-  ( z  =  (/)  ->  ( ( H `  z )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
1615rspcev 2997 . . 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 7533 . . 3  |-  om  e.  _V
19 cardinfima 7913 . . 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 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   _Vcvv 2901    u. cun 3263    C_ wss 3265   (/)c0 3573   U.cuni 3959   omcom 4787   ran crn 4821    |` cres 4822   "cima 4823    Fn wfn 5391   -->wf 5392   ` cfv 5396   reccrdg 6605   alephcale 7758
This theorem is referenced by:  alephfp  7924  alephfp2  7925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-har 7461  df-card 7761  df-aleph 7762
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