Theorem upgr1wlkdlem1 27005

Description: Lemma 1 for upgr1wlkd 27007. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
upgr1wlkd.p	⊢ 𝑃 = ⟨“𝑋𝑌”⟩
upgr1wlkd.f	⊢ 𝐹 = ⟨“𝐽”⟩
upgr1wlkd.x	⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺))
upgr1wlkd.y	⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺))
upgr1wlkd.j	⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})

Assertion

Ref	Expression
upgr1wlkdlem1	⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Proof of Theorem upgr1wlkdlem1

Step	Hyp	Ref	Expression
1		upgr1wlkd.j	. . 3 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
2		preq2 4269	. . . . . . 7 ⊢ (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
3	2	eqeq2d 2632	. . . . . 6 ⊢ (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
4	3	eqcoms 2630	. . . . 5 ⊢ (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
5		simpl 473	. . . . . . 7 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋})
6		dfsn2 4190	. . . . . . 7 ⊢ {𝑋} = {𝑋, 𝑋}
7	5, 6	syl6eqr 2674	. . . . . 6 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
8	7	ex 450	. . . . 5 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
9	4, 8	syl6bi 243	. . . 4 ⊢ (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋})))
10	9	com13 88	. . 3 ⊢ (𝜑 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝑋 = 𝑌 → ((iEdg‘𝐺)‘𝐽) = {𝑋})))
11	1, 10	mpd 15	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
12	11	imp 445	1 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177  {cpr 4179  ‘cfv 5888  ⟨“cs1 13294  ⟨“cs2 13586  Vtxcvtx 25874  iEdgciedg 25875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  upgr1wlkd  27007  upgr1trld  27008  upgr1pthd  27009  upgr1pthond  27010