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Theorem fin1a2lem1 8797
Description: Lemma for fin1a2 8812. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.a  |-  S  =  ( x  e.  On  |->  suc  x )
Assertion
Ref Expression
fin1a2lem1  |-  ( A  e.  On  ->  ( S `  A )  =  suc  A )

Proof of Theorem fin1a2lem1
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 suceloni 6647 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
2 suceq 4952 . . 3  |-  ( a  =  A  ->  suc  a  =  suc  A )
3 fin1a2lem.a . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
4 suceq 4952 . . . . 5  |-  ( x  =  a  ->  suc  x  =  suc  a )
54cbvmptv 4548 . . . 4  |-  ( x  e.  On  |->  suc  x
)  =  ( a  e.  On  |->  suc  a
)
63, 5eqtri 2486 . . 3  |-  S  =  ( a  e.  On  |->  suc  a )
72, 6fvmptg 5954 . 2  |-  ( ( A  e.  On  /\  suc  A  e.  On )  ->  ( S `  A )  =  suc  A )
81, 7mpdan 668 1  |-  ( A  e.  On  ->  ( S `  A )  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819    |-> cmpt 4515   Oncon0 4887   suc csuc 4889   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602
This theorem is referenced by:  fin1a2lem2  8798  fin1a2lem6  8802
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