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Theorem upgr1wlkdlem1 27974
 Description: Lemma 1 for upgr1wlkd 27976. (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 4633 . . . . . . 7 (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
32eqeq2d 2809 . . . . . 6 (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
43eqcoms 2806 . . . . 5 (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
5 simpl 486 . . . . . . 7 ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})
6 dfsn2 4541 . . . . . . 7 {𝑋} = {𝑋, 𝑋}
75, 6eqtr4di 2851 . . . . . 6 ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
87ex 416 . . . . 5 (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
94, 8syl6bi 256 . . . 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 410 1 ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4528  {cpr 4530  ‘cfv 6332  ⟨“cs1 13960  ⟨“cs2 14214  Vtxcvtx 26833  iEdgciedg 26834 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-un 3888  df-sn 4529  df-pr 4531 This theorem is referenced by:  upgr1wlkd  27976  upgr1trld  27977  upgr1pthd  27978  upgr1pthond  27979
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