Theorem wkslem1 27974

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 6774	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵))
2		fvoveq1 7298	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
3	1, 2	eqeq12d 2754	. 2 ⊢ (𝐴 = 𝐵 → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘(𝐵 + 1))))
4		2fveq3 6779	. . 3 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
5	1	sneqd 4573	. . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)})
6	4, 5	eqeq12d 2754	. 2 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
7	1, 2	preq12d 4677	. . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))})
8	7, 4	sseq12d 3954	. 2 ⊢ (𝐴 = 𝐵 → ({(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴)) ↔ {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))
9	3, 6, 8	ifpbi123d 1077	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)))))

Syntax hints:   → wi 4   ↔ wb 205  if-wif 1060   = wceq 1539   ⊆ wss 3887  {csn 4561  {cpr 4563  ‘cfv 6433  (class class class)co 7275  1c1 10872   + caddc 10874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  wlk1walk  28006  wlkres  28038  wlkp1lem8  28048  crctcshwlkn0lem6  28180  crctcshwlkn0lem7  28181  crctcshwlkn0  28186  pfxwlk  33085  revwlk  33086