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Theorem wkslem1 26559
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 6229 . . 3 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
2 oveq1 6697 . . . 4 (𝐴 = 𝐵 → (𝐴 + 1) = (𝐵 + 1))
32fveq2d 6233 . . 3 (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
41, 3eqeq12d 2666 . 2 (𝐴 = 𝐵 → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃‘(𝐵 + 1))))
5 fveq2 6229 . . . 4 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
65fveq2d 6233 . . 3 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
71sneqd 4222 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
86, 7eqeq12d 2666 . 2 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
91, 3preq12d 4308 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃‘(𝐵 + 1))})
109, 6sseq12d 3667 . 2 (𝐴 = 𝐵 → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵))))
114, 8, 10ifpbi123d 1047 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 196  if-wif 1032   = wceq 1523  wss 3607  {csn 4210  {cpr 4212  cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  wlk1walk  26591  wlkres  26623  crctcshwlkn0lem6  26763  crctcshwlkn0lem7  26764  crctcshwlkn0  26769
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