Theorem wkslem2 29543

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 6861	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵))
2	1	adantr 480	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘𝐴) = (𝑃‘𝐵))
3		fveq2 6861	. . . 4 ⊢ ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶))
4	3	adantl 481	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃‘𝐶))
5	2, 4	eqeq12d 2746	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘𝐶)))
6		2fveq3 6866	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
7	1	sneqd 4604	. . . 4 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)})
8	6, 7	eqeq12d 2746	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
9	8	adantr 480	. 2 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
10	2, 4	preq12d 4708	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘𝐶)})
11	6	adantr 480	. . 3 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
12	10, 11	sseq12d 3983	. 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 1078	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 206   ∧ wa 395  if-wif 1062   = wceq 1540   ⊆ wss 3917  {csn 4592  {cpr 4594  ‘cfv 6514  (class class class)co 7390  1c1 11076   + caddc 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522

This theorem is referenced by:  wlkl1loop  29573  wlk1walk  29574  crctcshwlkn0lem6  29752  1wlkdlem4  30076