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Theorem alephfplem1 7731
Description: Lemma for alephfp 7735. (Contributed by NM, 6-Nov-2004.)
Hypothesis
Ref Expression
alephfplem.1  |-  H  =  ( rec ( aleph ,  om )  |`  om )
Assertion
Ref Expression
alephfplem1  |-  ( H `
 (/) )  e.  ran  aleph

Proof of Theorem alephfplem1
StepHypRef Expression
1 omex 7344 . . . 4  |-  om  e.  _V
2 fr0g 6448 . . . 4  |-  ( om  e.  _V  ->  (
( rec ( aleph ,  om )  |`  om ) `  (/) )  =  om )
31, 2ax-mp 8 . . 3  |-  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )  =  om
4 alephfplem.1 . . . 4  |-  H  =  ( rec ( aleph ,  om )  |`  om )
54fveq1i 5526 . . 3  |-  ( H `
 (/) )  =  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )
6 aleph0 7693 . . 3  |-  ( aleph `  (/) )  =  om
73, 5, 63eqtr4i 2313 . 2  |-  ( H `
 (/) )  =  (
aleph `  (/) )
8 alephfnon 7692 . . 3  |-  aleph  Fn  On
9 0elon 4445 . . 3  |-  (/)  e.  On
10 fnfvelrn 5662 . . 3  |-  ( (
aleph  Fn  On  /\  (/)  e.  On )  ->  ( aleph `  (/) )  e. 
ran  aleph )
118, 9, 10mp2an 653 . 2  |-  ( aleph `  (/) )  e.  ran  aleph
127, 11eqeltri 2353 1  |-  ( H `
 (/) )  e.  ran  aleph
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   Oncon0 4392   omcom 4656   ran crn 4690    |` cres 4691    Fn wfn 5250   ` cfv 5255   reccrdg 6422   alephcale 7569
This theorem is referenced by:  alephfplem3  7733  alephfplem4  7734
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-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-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-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-recs 6388  df-rdg 6423  df-aleph 7573
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