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Theorem alephfplem1 7821
Description: Lemma for alephfp 7825. (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 7434 . . . 4  |-  om  e.  _V
2 fr0g 6535 . . . 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 5609 . . 3  |-  ( H `
 (/) )  =  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )
6 aleph0 7783 . . 3  |-  ( aleph `  (/) )  =  om
73, 5, 63eqtr4i 2388 . 2  |-  ( H `
 (/) )  =  (
aleph `  (/) )
8 alephfnon 7782 . . 3  |-  aleph  Fn  On
9 0elon 4527 . . 3  |-  (/)  e.  On
10 fnfvelrn 5745 . . 3  |-  ( (
aleph  Fn  On  /\  (/)  e.  On )  ->  ( aleph `  (/) )  e. 
ran  aleph )
118, 9, 10mp2an 653 . 2  |-  ( aleph `  (/) )  e.  ran  aleph
127, 11eqeltri 2428 1  |-  ( H `
 (/) )  e.  ran  aleph
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   Oncon0 4474   omcom 4738   ran crn 4772    |` cres 4773    Fn wfn 5332   ` cfv 5337   reccrdg 6509   alephcale 7659
This theorem is referenced by:  alephfplem3  7823  alephfplem4  7824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-aleph 7663
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