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Theorem upgr11wlkdlem1 41312
 Description: Lemma 1 for upgr11wlkd 41314. (Contributed by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
upgr11wlkd.p 𝑃 = ⟨“𝑋𝑌”⟩
upgr11wlkd.f 𝐹 = ⟨“𝐽”⟩
upgr11wlkd.x (𝜑𝑋 ∈ (Vtx‘𝐺))
upgr11wlkd.y (𝜑𝑌 ∈ (Vtx‘𝐺))
upgr11wlkd.j (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
Assertion
Ref Expression
upgr11wlkdlem1 ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Proof of Theorem upgr11wlkdlem1
StepHypRef Expression
1 upgr11wlkd.j . . 3 (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
2 preq2 4213 . . . . . . 7 (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
32eqeq2d 2620 . . . . . 6 (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
43eqcoms 2618 . . . . 5 (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
5 simpl 472 . . . . . . 7 ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})
6 dfsn2 4138 . . . . . . 7 {𝑋} = {𝑋, 𝑋}
75, 6syl6eqr 2662 . . . . . 6 ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
87ex 449 . . . . 5 (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
94, 8syl6bi 242 . . . 4 (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋})))
109com13 86 . . 3 (𝜑 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝑋 = 𝑌 → ((iEdg‘𝐺)‘𝐽) = {𝑋})))
111, 10mpd 15 . 2 (𝜑 → (𝑋 = 𝑌 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
1211imp 444 1 ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  {cpr 4127  ‘cfv 5804  ⟨“cs1 13149  ⟨“cs2 13437  Vtxcvtx 25673  iEdgciedg 25674 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-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-sn 4126  df-pr 4128 This theorem is referenced by:  upgr11wlkd  41314  upgr1trld  41315  upgr1pthd  41316  upgr1pthond  41317
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