MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem1 Structured version   Unicode version

Theorem fin1a2lem1 8285
Description: Lemma for fin1a2 8300. (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 4796 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
2 suceq 4649 . . 3  |-  ( a  =  A  ->  suc  a  =  suc  A )
3 fin1a2lem.a . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
4 suceq 4649 . . . . 5  |-  ( x  =  a  ->  suc  x  =  suc  a )
54cbvmptv 4303 . . . 4  |-  ( x  e.  On  |->  suc  x
)  =  ( a  e.  On  |->  suc  a
)
63, 5eqtri 2458 . . 3  |-  S  =  ( a  e.  On  |->  suc  a )
72, 6fvmptg 5807 . 2  |-  ( ( A  e.  On  /\  suc  A  e.  On )  ->  ( S `  A )  =  suc  A )
81, 7mpdan 651 1  |-  ( A  e.  On  ->  ( S `  A )  =  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    e. cmpt 4269   Oncon0 4584   suc csuc 4586   ` cfv 5457
This theorem is referenced by:  fin1a2lem2  8286  fin1a2lem6  8290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465
  Copyright terms: Public domain W3C validator