Theorem wkslem2 27382

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))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵)))))

Proof of Theorem wkslem2

Step	Hyp	Ref	Expression
1		fveq2 6663	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵))
2	1	adantr 483	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘𝐴) = (𝑃‘𝐵))
3		fveq2 6663	. . . 4 ⊢ ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶))
4	3	adantl 484	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶))
5	2, 4	eqeq12d 2835	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘𝐶)))
6		2fveq3 6668	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
7	1	sneqd 4571	. . . 4 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)})
8	6, 7	eqeq12d 2835	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
9	8	adantr 483	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
10	2, 4	preq12d 4669	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘𝐶)})
11	6	adantr 483	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
12	10, 11	sseq12d 3998	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ({(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴)) ↔ {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵))))
13	5, 9, 12	ifpbi123d 1072	1 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  if-wif 1057   = wceq 1531   ⊆ wss 3934  {csn 4559  {cpr 4561  ‘cfv 6348  (class class class)co 7148  1c1 10530   + caddc 10532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356

This theorem is referenced by:  wlkl1loop  27411  wlk1walk  27412  crctcshwlkn0lem6  27585  1wlkdlem4  27911