Theorem wkslem2 16171

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 5639	. . . 4 |- ( A = B -> ( P ` A ) = ( P ` B ) )
2	1	adantr 276	. . 3 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( P ` A ) = ( P ` B ) )
3		fveq2 5639	. . . 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 2246	. 2 |- ( ( A = B /\ ( A + 1 ) = C ) -> ( ( P ` A ) = ( P ` ( A + 1 ) ) <-> ( P ` B ) = ( P ` C ) ) )
6		2fveq3 5644	. . . 4 |- ( A = B -> ( I ` ( F ` A ) ) = ( I ` ( F ` B ) ) )
7	1	sneqd 3682	. . . 4 |- ( A = B -> { ( P ` A ) } = { ( P ` B ) } )
8	6, 7	eqeq12d 2246	. . 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 3756	. . 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 3258	. 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 1000	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 985   = wceq 1397   C_ wss 3200   {csn 3669   {cpr 3670   ` cfv 5326  (class class class)co 6017   1c1 8032   + caddc 8034

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-ifp 986  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334

This theorem is referenced by:  wlkl1loop  16208  wlk1walkdom  16209