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Theorem alephfplem2 7912
Description: Lemma for alephfp 7915. (Contributed by NM, 6-Nov-2004.)
Hypothesis
Ref Expression
alephfplem.1  |-  H  =  ( rec ( aleph ,  om )  |`  om )
Assertion
Ref Expression
alephfplem2  |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w )
) )
Distinct variable group:    w, H

Proof of Theorem alephfplem2
StepHypRef Expression
1 frsuc 6623 . 2  |-  ( w  e.  om  ->  (
( rec ( aleph ,  om )  |`  om ) `  suc  w )  =  ( aleph `  ( ( rec ( aleph ,  om )  |` 
om ) `  w
) ) )
2 alephfplem.1 . . 3  |-  H  =  ( rec ( aleph ,  om )  |`  om )
32fveq1i 5662 . 2  |-  ( H `
 suc  w )  =  ( ( rec ( aleph ,  om )  |` 
om ) `  suc  w )
42fveq1i 5662 . . 3  |-  ( H `
 w )  =  ( ( rec ( aleph ,  om )  |`  om ) `  w )
54fveq2i 5664 . 2  |-  ( aleph `  ( H `  w
) )  =  (
aleph `  ( ( rec ( aleph ,  om )  |` 
om ) `  w
) )
61, 3, 53eqtr4g 2437 1  |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   suc csuc 4517   omcom 4778    |` cres 4813   ` cfv 5387   reccrdg 6596   alephcale 7749
This theorem is referenced by:  alephfplem3  7913  alephfp  7915
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-recs 6562  df-rdg 6597
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