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Theorem upgr1wlkdlem1 27123
Description: Lemma 1 for upgr1wlkd 27125. (Contributed by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
upgr1wlkd.p 𝑃 = ⟨“𝑋𝑌”⟩
upgr1wlkd.f 𝐹 = ⟨“𝐽”⟩
upgr1wlkd.x (𝜑𝑋 ∈ (Vtx‘𝐺))
upgr1wlkd.y (𝜑𝑌 ∈ (Vtx‘𝐺))
upgr1wlkd.j (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
Assertion
Ref Expression
upgr1wlkdlem1 ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Proof of Theorem upgr1wlkdlem1
StepHypRef Expression
1 upgr1wlkd.j . . 3 (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
2 preq2 4301 . . . . . . 7 (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
32eqeq2d 2661 . . . . . 6 (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
43eqcoms 2659 . . . . 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 4223 . . . . . . 7 {𝑋} = {𝑋, 𝑋}
75, 6syl6eqr 2703 . . . . . 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 243 . . . 4 (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋})))
109com13 88 . . 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 196  wa 383   = wceq 1523  wcel 2030  {csn 4210  {cpr 4212  cfv 5926  ⟨“cs1 13326  ⟨“cs2 13632  Vtxcvtx 25919  iEdgciedg 25920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213
This theorem is referenced by:  upgr1wlkd  27125  upgr1trld  27126  upgr1pthd  27127  upgr1pthond  27128
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