Theorem wkslem1 30015

Description: Lemma 1 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
wkslem1	⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵)))))

Proof of Theorem wkslem1

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵))
2		fvoveq1 7442	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
3	1, 2	eqeq12d 2781	. 2 ⊢ (𝐴 = 𝐵 → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘(𝐵 + 1))))
4		2fveq3 6890	. . 3 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
5	1	sneqd 4603	. . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)})
6	4, 5	eqeq12d 2781	. 2 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
7	1, 2	preq12d 4709	. . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))})
8	7, 4	sseq12d 3971	. 2 ⊢ (𝐴 = 𝐵 → ({(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴)) ↔ {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))
9	3, 6, 8	ifpbi123d 1095	1 ⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  if-wif 1078   = wceq 1570   ⊆ wss 3906  {csn 4591  {cpr 4593  ‘cfv 6540  (class class class)co 7419  1c1 11116   + caddc 11118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  wlk1walk  30046  wlkres  30076  wlkp1lem8  30086  pfxwlk  30093  revwlk  30094  crctcshwlkn0lem6  30231  crctcshwlkn0lem7  30232  crctcshwlkn0  30237