Theorem wkslem1 26503

Description: Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)

Assertion

Ref	Expression
wkslem1	|- ( A = B -> (if- ( ( P ` A ) = ( P ` ( A + 1 ) ) , ( I ` ( F ` A ) ) = { ( P ` A ) } , { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) ) <-> if- ( ( P ` B ) = ( P ` ( B + 1 ) ) , ( I ` ( F ` B ) ) = { ( P ` B ) } , { ( P ` B ) , ( P ` ( B + 1 ) ) } C_ ( I ` ( F ` B ) ) ) ) )

Proof of Theorem wkslem1

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( A = B -> ( P ` A ) = ( P ` B ) )
2		oveq1 6657	. . . 4 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
3	2	fveq2d 6195	. . 3 |- ( A = B -> ( P ` ( A + 1 ) ) = ( P ` ( B + 1 ) ) )
4	1, 3	eqeq12d 2637	. 2 |- ( A = B -> ( ( P ` A ) = ( P ` ( A + 1 ) ) <-> ( P ` B ) = ( P ` ( B + 1 ) ) ) )
5		fveq2 6191	. . . 4 |- ( A = B -> ( F ` A ) = ( F ` B ) )
6	5	fveq2d 6195	. . 3 |- ( A = B -> ( I ` ( F ` A ) ) = ( I ` ( F ` B ) ) )
7	1	sneqd 4189	. . 3 |- ( A = B -> { ( P ` A ) } = { ( P ` B ) } )
8	6, 7	eqeq12d 2637	. 2 |- ( A = B -> ( ( I ` ( F ` A ) ) = { ( P ` A ) } <-> ( I ` ( F ` B ) ) = { ( P ` B ) } ) )
9	1, 3	preq12d 4276	. . 3 |- ( A = B -> { ( P ` A ) , ( P ` ( A + 1 ) ) } = { ( P ` B ) , ( P ` ( B + 1 ) ) } )
10	9, 6	sseq12d 3634	. 2 |- ( A = B -> ( { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) <-> { ( P ` B ) , ( P ` ( B + 1 ) ) } C_ ( I ` ( F ` B ) ) ) )
11	4, 8, 10	ifpbi123d 1027	1 |- ( A = B -> (if- ( ( P ` A ) = ( P ` ( A + 1 ) ) , ( I ` ( F ` A ) ) = { ( P ` A ) } , { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) ) <-> if- ( ( P ` B ) = ( P ` ( B + 1 ) ) , ( I ` ( F ` B ) ) = { ( P ` B ) } , { ( P ` B ) , ( P ` ( B + 1 ) ) } C_ ( I ` ( F ` B ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196  if-wif 1012   = wceq 1483   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  wlk1walk  26535  wlkres  26567  crctcshwlkn0lem6  26707  crctcshwlkn0lem7  26708  crctcshwlkn0  26713