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Theorem wkslem2 27497
 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 6658 . . . 4 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
21adantr 484 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃𝐴) = (𝑃𝐵))
3 fveq2 6658 . . . 4 ((𝐴 + 1) = 𝐶 → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
43adantl 485 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝑃‘(𝐴 + 1)) = (𝑃𝐶))
52, 4eqeq12d 2774 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃𝐶)))
6 2fveq3 6663 . . . 4 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
71sneqd 4534 . . . 4 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
86, 7eqeq12d 2774 . . 3 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
98adantr 484 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
102, 4preq12d 4634 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃𝐶)})
116adantr 484 . . 3 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
1210, 11sseq12d 3925 . 2 ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵))))
135, 9, 12ifpbi123d 1075 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 209   ∧ wa 399  if-wif 1058   = wceq 1538   ⊆ wss 3858  {csn 4522  {cpr 4524  ‘cfv 6335  (class class class)co 7150  1c1 10576   + caddc 10578 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343 This theorem is referenced by:  wlkl1loop  27526  wlk1walk  27527  crctcshwlkn0lem6  27700  1wlkdlem4  28024
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