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Theorem wkslem2 26739
Description: Lemma 2 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
wkslem2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃𝐶), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵)))))

Proof of Theorem wkslem2
StepHypRef Expression
1 fveq2 6332 . . . 4 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
21adantr 466 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃𝐴) = (𝑃𝐵))
3 fveq2 6332 . . . 4 ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
43adantl 467 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
52, 4eqeq12d 2786 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃𝐶)))
6 fveq2 6332 . . . . 5 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
76fveq2d 6336 . . . 4 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
81sneqd 4328 . . . 4 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
97, 8eqeq12d 2786 . . 3 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
109adantr 466 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
112, 4preq12d 4412 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃𝐶)})
127adantr 466 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
1311, 12sseq12d 3783 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵))))
145, 10, 13ifpbi123d 1064 1 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃𝐶), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  if-wif 1049   = wceq 1631  wss 3723  {csn 4316  {cpr 4318  cfv 6031  (class class class)co 6793  1c1 10139   + caddc 10141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ifp 1050  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039
This theorem is referenced by:  wlkl1loop  26769  wlk1walk  26770  crctcshwlkn0lem6  26943  1wlkdlem4  27320
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