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Theorem wkslem1 27401
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 6649 . . 3 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
2 fvoveq1 7162 . . 3 (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
31, 2eqeq12d 2817 . 2 (𝐴 = 𝐵 → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃‘(𝐵 + 1))))
4 2fveq3 6654 . . 3 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
51sneqd 4540 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
64, 5eqeq12d 2817 . 2 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
71, 2preq12d 4640 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃‘(𝐵 + 1))})
87, 4sseq12d 3951 . 2 (𝐴 = 𝐵 → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵))))
93, 6, 8ifpbi123d 1075 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 209  if-wif 1058   = wceq 1538  wss 3884  {csn 4528  {cpr 4530  cfv 6328  (class class class)co 7139  1c1 10531   + caddc 10533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142
This theorem is referenced by:  wlk1walk  27432  wlkres  27464  wlkp1lem8  27474  crctcshwlkn0lem6  27605  crctcshwlkn0lem7  27606  crctcshwlkn0  27611  pfxwlk  32484  revwlk  32485
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