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Theorem wkslem2 28264
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 6825 . . . 4 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
21adantr 481 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃𝐴) = (𝑃𝐵))
3 fveq2 6825 . . . 4 ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
43adantl 482 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
52, 4eqeq12d 2752 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃𝐶)))
6 2fveq3 6830 . . . 4 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
71sneqd 4585 . . . 4 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
86, 7eqeq12d 2752 . . 3 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
98adantr 481 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
102, 4preq12d 4689 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃𝐶)})
116adantr 481 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
1210, 11sseq12d 3965 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵))))
135, 9, 12ifpbi123d 1077 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 205  wa 396  if-wif 1060   = wceq 1540  wss 3898  {csn 4573  {cpr 4575  cfv 6479  (class class class)co 7337  1c1 10973   + caddc 10975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487
This theorem is referenced by:  wlkl1loop  28294  wlk1walk  28295  crctcshwlkn0lem6  28468  1wlkdlem4  28792
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