Theorem wkslem2 16316

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 5670	. . . 4 |- ( A = B -> ( P ` A ) = ( P ` B ) )
2	1	adantr 276	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( P ` A ) = ( P ` B ) )
3		fveq2 5670	. . . 4 |- ( ( A + 1 ) = C -> ( P ` ( A + 1 ) ) = ( P ` C ) )
4	3	adantl 277	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( P ` ( A + 1 ) ) = ( P ` C ) )
5	2, 4	eqeq12d 2247	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( ( P ` A ) = ( P ` ( A + 1 ) ) <-> ( P ` B ) = ( P ` C ) ) )
6		2fveq3 5675	. . . 4 |- ( A = B -> ( I ` ( F ` A ) ) = ( I ` ( F ` B ) ) )
7	1	sneqd 3702	. . . 4 |- ( A = B -> { ( P ` A ) } = { ( P ` B ) } )
8	6, 7	eqeq12d 2247	. . 3 |- ( A = B -> ( ( I ` ( F ` A ) ) = { ( P ` A ) } <-> ( I ` ( F ` B ) ) = { ( P ` B ) } ) )
9	8	adantr 276	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( ( I ` ( F ` A ) ) = { ( P ` A ) } <-> ( I ` ( F ` B ) ) = { ( P ` B ) } ) )
10	2, 4	preq12d 3776	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> { ( P ` A ) , ( P ` ( A + 1 ) ) } = { ( P ` B ) , ( P ` C ) } )
11	6	adantr 276	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( I ` ( F ` A ) ) = ( I ` ( F ` B ) ) )
12	10, 11	sseq12d 3269	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) <-> { ( P ` B ) , ( P ` C ) } C_ ( I ` ( F ` B ) ) ) )
13	5, 9, 12	ifpbi123d 1001	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   /\ wa 104   <-> wb 105  if-wif 986   = wceq 1398   C_ wss 3211   {csn 3689   {cpr 3690   ` cfv 5352  (class class class)co 6050   1c1 8128   + caddc 8130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-ifp 987  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360

This theorem is referenced by:  wlkl1loop  16353  wlk1walkdom  16354