Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1wlkslem1 Structured version   Visualization version   GIF version

Theorem 1wlkslem1 40809
 Description: Lemma 1 for 1-walks to substitute the index of the condition for vertices and edges in a 1-walk. (Contributed by AV, 23-Apr-2021.)
Assertion
Ref Expression
1wlkslem1 (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 1wlkslem1
StepHypRef Expression
1 fveq2 6103 . . 3 (𝐴 = 𝐵 → (𝑃𝐴) = (𝑃𝐵))
2 oveq1 6556 . . . 4 (𝐴 = 𝐵 → (𝐴 + 1) = (𝐵 + 1))
32fveq2d 6107 . . 3 (𝐴 = 𝐵 → (𝑃‘(𝐴 + 1)) = (𝑃‘(𝐵 + 1)))
41, 3eqeq12d 2625 . 2 (𝐴 = 𝐵 → ((𝑃𝐴) = (𝑃‘(𝐴 + 1)) ↔ (𝑃𝐵) = (𝑃‘(𝐵 + 1))))
5 fveq2 6103 . . . 4 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
65fveq2d 6107 . . 3 (𝐴 = 𝐵 → (𝐼‘(𝐹𝐴)) = (𝐼‘(𝐹𝐵)))
71sneqd 4137 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴)} = {(𝑃𝐵)})
86, 7eqeq12d 2625 . 2 (𝐴 = 𝐵 → ((𝐼‘(𝐹𝐴)) = {(𝑃𝐴)} ↔ (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}))
91, 3preq12d 4220 . . 3 (𝐴 = 𝐵 → {(𝑃𝐴), (𝑃‘(𝐴 + 1))} = {(𝑃𝐵), (𝑃‘(𝐵 + 1))})
109, 6sseq12d 3597 . 2 (𝐴 = 𝐵 → ({(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴)) ↔ {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵))))
114, 8, 10ifpbi123d 1021 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 195  if-wif 1006   = wceq 1475   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  1wlk1walk  40843  1wlkres  40879  crctcsh1wlkn0lem6  41018  crctcsh1wlkn0lem7  41019  crctcsh1wlkn0  41024
 Copyright terms: Public domain W3C validator