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Theorem fin1a2lem1 8781
Description: Lemma for fin1a2 8796. (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 6633 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
2 suceq 4943 . . 3  |-  ( a  =  A  ->  suc  a  =  suc  A )
3 fin1a2lem.a . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
4 suceq 4943 . . . . 5  |-  ( x  =  a  ->  suc  x  =  suc  a )
54cbvmptv 4538 . . . 4  |-  ( x  e.  On  |->  suc  x
)  =  ( a  e.  On  |->  suc  a
)
63, 5eqtri 2496 . . 3  |-  S  =  ( a  e.  On  |->  suc  a )
72, 6fvmptg 5949 . 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 1379    e. wcel 1767    |-> cmpt 4505   Oncon0 4878   suc csuc 4880   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  fin1a2lem2  8782  fin1a2lem6  8786
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