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Theorem 1wlkvtxeledglem 40827
 Description: Lemma for 1wlkvtxeledg 40828: Two adjacent vertices in a 1-walk are incident with an edge. (Contributed by AV, 4-Apr-2021.)
Assertion
Ref Expression
1wlkvtxeledglem (if-((𝑃𝐾) = (𝑃‘(𝐾 + 1)), (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}, {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾))) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾)))

Proof of Theorem 1wlkvtxeledglem
StepHypRef Expression
1 ssid 3587 . . . 4 {(𝑃𝐾)} ⊆ {(𝑃𝐾)}
21a1i 11 . . 3 (((𝑃𝐾) = (𝑃‘(𝐾 + 1)) ∧ (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}) → {(𝑃𝐾)} ⊆ {(𝑃𝐾)})
3 preq2 4213 . . . . . 6 ((𝑃‘(𝐾 + 1)) = (𝑃𝐾) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} = {(𝑃𝐾), (𝑃𝐾)})
4 dfsn2 4138 . . . . . 6 {(𝑃𝐾)} = {(𝑃𝐾), (𝑃𝐾)}
53, 4syl6eqr 2662 . . . . 5 ((𝑃‘(𝐾 + 1)) = (𝑃𝐾) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} = {(𝑃𝐾)})
65eqcoms 2618 . . . 4 ((𝑃𝐾) = (𝑃‘(𝐾 + 1)) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} = {(𝑃𝐾)})
76adantr 480 . . 3 (((𝑃𝐾) = (𝑃‘(𝐾 + 1)) ∧ (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} = {(𝑃𝐾)})
8 simpr 476 . . 3 (((𝑃𝐾) = (𝑃‘(𝐾 + 1)) ∧ (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}) → (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)})
92, 7, 83sstr4d 3611 . 2 (((𝑃𝐾) = (𝑃‘(𝐾 + 1)) ∧ (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾)))
1091fpid3 1023 1 (if-((𝑃𝐾) = (𝑃‘(𝐾 + 1)), (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}, {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾))) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  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-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128 This theorem is referenced by:  1wlkvtxeledg  40828
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