Theorem wkslem2 26504

Description: Lemma 2 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
wkslem2	|- ( ( A = B /\ ( A + 1 ) = C ) -> (if- ( ( P ` A ) = ( P ` ( A + 1 ) ) , ( I ` ( F ` A ) ) = { ( P ` A ) } , { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) ) <-> if- ( ( P ` B ) = ( P ` C ) , ( I ` ( F ` B ) ) = { ( P ` B ) } , { ( P ` B ) , ( P ` C ) } C_ ( I ` ( F ` B ) ) ) ) )

Proof of Theorem wkslem2

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( A = B -> ( P ` A ) = ( P ` B ) )
2	1	adantr 481	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( P ` A ) = ( P ` B ) )
3		fveq2 6191	. . . 4 |- ( ( A + 1 ) = C -> ( P ` ( A + 1 ) ) = ( P ` C ) )
4	3	adantl 482	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( P ` ( A + 1 ) ) = ( P ` C ) )
5	2, 4	eqeq12d 2637	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( ( P ` A ) = ( P ` ( A + 1 ) ) <-> ( P ` B ) = ( P ` C ) ) )
6		fveq2 6191	. . . . 5 |- ( A = B -> ( F ` A ) = ( F ` B ) )
7	6	fveq2d 6195	. . . 4 |- ( A = B -> ( I ` ( F ` A ) ) = ( I ` ( F ` B ) ) )
8	1	sneqd 4189	. . . 4 |- ( A = B -> { ( P ` A ) } = { ( P ` B ) } )
9	7, 8	eqeq12d 2637	. . 3 |- ( A = B -> ( ( I ` ( F ` A ) ) = { ( P ` A ) } <-> ( I ` ( F ` B ) ) = { ( P ` B ) } ) )
10	9	adantr 481	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( ( I ` ( F ` A ) ) = { ( P ` A ) } <-> ( I ` ( F ` B ) ) = { ( P ` B ) } ) )
11	2, 4	preq12d 4276	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> { ( P ` A ) , ( P ` ( A + 1 ) ) } = { ( P ` B ) , ( P ` C ) } )
12	7	adantr 481	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( I ` ( F ` A ) ) = ( I ` ( F ` B ) ) )
13	11, 12	sseq12d 3634	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) <-> { ( P ` B ) , ( P ` C ) } C_ ( I ` ( F ` B ) ) ) )
14	5, 10, 13	ifpbi123d 1027	1 |- ( ( A = B /\ ( A + 1 ) = C ) -> (if- ( ( P ` A ) = ( P ` ( A + 1 ) ) , ( I ` ( F ` A ) ) = { ( P ` A ) } , { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) ) <-> if- ( ( P ` B ) = ( P ` C ) , ( I ` ( F ` B ) ) = { ( P ` B ) } , { ( P ` B ) , ( P ` C ) } C_ ( I ` ( F ` B ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384  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

This theorem is referenced by:  wlkl1loop  26534  wlk1walk  26535  crctcshwlkn0lem6  26707  1wlkdlem4  27000