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Theorem alephfplem1 8023
Description: Lemma for alephfp 8027. (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 7634 . . . 4  |-  om  e.  _V
2 fr0g 6729 . . . 4  |-  ( om  e.  _V  ->  (
( rec ( aleph ,  om )  |`  om ) `  (/) )  =  om )
31, 2ax-mp 5 . . 3  |-  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )  =  om
4 alephfplem.1 . . . 4  |-  H  =  ( rec ( aleph ,  om )  |`  om )
54fveq1i 5764 . . 3  |-  ( H `
 (/) )  =  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )
6 aleph0 7985 . . 3  |-  ( aleph `  (/) )  =  om
73, 5, 63eqtr4i 2473 . 2  |-  ( H `
 (/) )  =  (
aleph `  (/) )
8 alephfnon 7984 . . 3  |-  aleph  Fn  On
9 0elon 4669 . . 3  |-  (/)  e.  On
10 fnfvelrn 5903 . . 3  |-  ( (
aleph  Fn  On  /\  (/)  e.  On )  ->  ( aleph `  (/) )  e. 
ran  aleph )
118, 9, 10mp2an 655 . 2  |-  ( aleph `  (/) )  e.  ran  aleph
127, 11eqeltri 2513 1  |-  ( H `
 (/) )  e.  ran  aleph
Colors of variables: wff set class
Syntax hints:    = wceq 1654    e. wcel 1728   _Vcvv 2965   (/)c0 3616   Oncon0 4616   omcom 4880   ran crn 4914    |` cres 4915    Fn wfn 5484   ` cfv 5489   reccrdg 6703   alephcale 7861
This theorem is referenced by:  alephfplem3  8025  alephfplem4  8026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-inf2 7632
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-recs 6669  df-rdg 6704  df-aleph 7865
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