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Theorem upgr1wlkdlem1 30174
Description: Lemma 1 for upgr1wlkd 30176. (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 4739 . . . . . . 7 (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
32eqeq2d 2746 . . . . . 6 (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
43eqcoms 2743 . . . . 5 (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
5 simpl 482 . . . . . . 7 ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})
6 dfsn2 4644 . . . . . . 7 {𝑋} = {𝑋, 𝑋}
75, 6eqtr4di 2793 . . . . . 6 ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
87ex 412 . . . . 5 (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
94, 8biimtrdi 253 . . . 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 406 1 ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {csn 4631  {cpr 4633  cfv 6563  ⟨“cs1 14630  ⟨“cs2 14877  Vtxcvtx 29028  iEdgciedg 29029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by:  upgr1wlkd  30176  upgr1trld  30177  upgr1pthd  30178  upgr1pthond  30179
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