Theorem upgr1wlkdlem1 30165

Description: Lemma 1 for upgr1wlkd 30167. (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 4733	. . . . . . 7 ⊢ (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
3	2	eqeq2d 2747	. . . . . 6 ⊢ (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
4	3	eqcoms 2744	. . . . 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 4638	. . . . . . 7 ⊢ {𝑋} = {𝑋, 𝑋}
7	5, 6	eqtr4di 2794	. . . . . 6 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
8	7	ex 412	. . . . 5 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
9	4, 8	biimtrdi 253	. . . 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 406	1 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {csn 4625  {cpr 4627  ‘cfv 6560  ⟨“cs1 14634  ⟨“cs2 14881  Vtxcvtx 29014  iEdgciedg 29015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-sn 4626  df-pr 4628

This theorem is referenced by:  upgr1wlkd  30167  upgr1trld  30168  upgr1pthd  30169  upgr1pthond  30170