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Theorem alephfplem1 7945
Description: Lemma for alephfp 7949. (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 7558 . . . 4  |-  om  e.  _V
2 fr0g 6656 . . . 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 5692 . . 3  |-  ( H `
 (/) )  =  ( ( rec ( aleph ,  om )  |`  om ) `  (/) )
6 aleph0 7907 . . 3  |-  ( aleph `  (/) )  =  om
73, 5, 63eqtr4i 2438 . 2  |-  ( H `
 (/) )  =  (
aleph `  (/) )
8 alephfnon 7906 . . 3  |-  aleph  Fn  On
9 0elon 4598 . . 3  |-  (/)  e.  On
10 fnfvelrn 5830 . . 3  |-  ( (
aleph  Fn  On  /\  (/)  e.  On )  ->  ( aleph `  (/) )  e. 
ran  aleph )
118, 9, 10mp2an 654 . 2  |-  ( aleph `  (/) )  e.  ran  aleph
127, 11eqeltri 2478 1  |-  ( H `
 (/) )  e.  ran  aleph
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2920   (/)c0 3592   Oncon0 4545   omcom 4808   ran crn 4842    |` cres 4843    Fn wfn 5412   ` cfv 5417   reccrdg 6630   alephcale 7783
This theorem is referenced by:  alephfplem3  7947  alephfplem4  7948
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-aleph 7787
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