Theorem wkslem1 29588

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 6840	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘𝐴) = (𝑃‘𝐵))
2		fvoveq1 7392	. . 3 ⊢ (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
3	1, 2	eqeq12d 2745	. 2 ⊢ (𝐴 = 𝐵 → ((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃‘𝐵) = (𝑃‘(𝐵 + 1))))
4		2fveq3 6845	. . 3 ⊢ (𝐴 = 𝐵 → (𝐼‘(𝐹‘𝐴)) = (𝐼‘(𝐹‘𝐵)))
5	1	sneqd 4597	. . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴)} = {(𝑃‘𝐵)})
6	4, 5	eqeq12d 2745	. 2 ⊢ (𝐴 = 𝐵 → ((𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)} ↔ (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}))
7	1, 2	preq12d 4701	. . 3 ⊢ (𝐴 = 𝐵 → {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))})
8	7, 4	sseq12d 3977	. 2 ⊢ (𝐴 = 𝐵 → ({(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴)) ↔ {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))
9	3, 6, 8	ifpbi123d 1078	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 206  if-wif 1062   = wceq 1540   ⊆ wss 3911  {csn 4585  {cpr 4587  ‘cfv 6499  (class class class)co 7369  1c1 11045   + caddc 11047

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372

This theorem is referenced by:  wlk1walk  29619  wlkres  29649  wlkp1lem8  29659  crctcshwlkn0lem6  29795  crctcshwlkn0lem7  29796  crctcshwlkn0  29801  pfxwlk  35104  revwlk  35105