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Theorem wkslem2 29641
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 6907 . . . 4 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
21adantr 480 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃𝐴) = (𝑃𝐵))
3 fveq2 6907 . . . 4 ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
43adantl 481 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
52, 4eqeq12d 2751 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃𝐶)))
6 2fveq3 6912 . . . 4 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
71sneqd 4643 . . . 4 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
86, 7eqeq12d 2751 . . 3 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
98adantr 480 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
102, 4preq12d 4746 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃𝐶)})
116adantr 480 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
1210, 11sseq12d 4029 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵))))
135, 9, 12ifpbi123d 1078 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 206  wa 395  if-wif 1062   = wceq 1537  wss 3963  {csn 4631  {cpr 4633  cfv 6563  (class class class)co 7431  1c1 11154   + caddc 11156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  wlkl1loop  29671  wlk1walk  29672  crctcshwlkn0lem6  29845  1wlkdlem4  30169
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