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Theorem wkslem1 28360
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-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵)))))

Proof of Theorem wkslem1
StepHypRef Expression
1 fveq2 6838 . . 3 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
2 fvoveq1 7373 . . 3 (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
31, 2eqeq12d 2754 . 2 (𝐴 = 𝐵 → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃‘(𝐵 + 1))))
4 2fveq3 6843 . . 3 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
51sneqd 4597 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
64, 5eqeq12d 2754 . 2 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
71, 2preq12d 4701 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃‘(𝐵 + 1))})
87, 4sseq12d 3976 . 2 (𝐴 = 𝐵 → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵))))
93, 6, 8ifpbi123d 1079 1 (𝐴 = 𝐵 → (if-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  if-wif 1062   = wceq 1542  wss 3909  {csn 4585  {cpr 4587  cfv 6492  (class class class)co 7350  1c1 10986   + caddc 10988
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6444  df-fv 6500  df-ov 7353
This theorem is referenced by:  wlk1walk  28392  wlkres  28423  wlkp1lem8  28433  crctcshwlkn0lem6  28565  crctcshwlkn0lem7  28566  crctcshwlkn0  28571  pfxwlk  33497  revwlk  33498
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