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Theorem alephfplem1 7941
Description: Lemma for alephfp 7945. (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 7554 . . . 4  |-  om  e.  _V
2 fr0g 6652 . . . 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 5688 . . 3  |-  ( H `
 (/) )  =  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )
6 aleph0 7903 . . 3  |-  ( aleph `  (/) )  =  om
73, 5, 63eqtr4i 2434 . 2  |-  ( H `
 (/) )  =  (
aleph `  (/) )
8 alephfnon 7902 . . 3  |-  aleph  Fn  On
9 0elon 4594 . . 3  |-  (/)  e.  On
10 fnfvelrn 5826 . . 3  |-  ( (
aleph  Fn  On  /\  (/)  e.  On )  ->  ( aleph `  (/) )  e. 
ran  aleph )
118, 9, 10mp2an 654 . 2  |-  ( aleph `  (/) )  e.  ran  aleph
127, 11eqeltri 2474 1  |-  ( H `
 (/) )  e.  ran  aleph
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   Oncon0 4541   omcom 4804   ran crn 4838    |` cres 4839    Fn wfn 5408   ` cfv 5413   reccrdg 6626   alephcale 7779
This theorem is referenced by:  alephfplem3  7943  alephfplem4  7944
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-aleph 7783
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