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

Theorem fin1a2lem1 8240
Description: Lemma for fin1a2 8255. (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 4756 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
2 suceq 4610 . . 3  |-  ( a  =  A  ->  suc  a  =  suc  A )
3 fin1a2lem.a . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
4 suceq 4610 . . . . 5  |-  ( x  =  a  ->  suc  x  =  suc  a )
54cbvmptv 4264 . . . 4  |-  ( x  e.  On  |->  suc  x
)  =  ( a  e.  On  |->  suc  a
)
63, 5eqtri 2428 . . 3  |-  S  =  ( a  e.  On  |->  suc  a )
72, 6fvmptg 5767 . 2  |-  ( ( A  e.  On  /\  suc  A  e.  On )  ->  ( S `  A )  =  suc  A )
81, 7mpdan 650 1  |-  ( A  e.  On  ->  ( S `  A )  =  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    e. cmpt 4230   Oncon0 4545   suc csuc 4547   ` cfv 5417
This theorem is referenced by:  fin1a2lem2  8241  fin1a2lem6  8245
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-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
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-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  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-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425
  Copyright terms: Public domain W3C validator