Theorem upgr11wlkdlem1 41312

Description: Lemma 1 for upgr11wlkd 41314. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
upgr11wlkd.p	⊢ 𝑃 = ⟨“𝑋𝑌”⟩
upgr11wlkd.f	⊢ 𝐹 = ⟨“𝐽”⟩
upgr11wlkd.x	⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺))
upgr11wlkd.y	⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺))
upgr11wlkd.j	⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})

Assertion

Ref	Expression
upgr11wlkdlem1	⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Proof of Theorem upgr11wlkdlem1

Step	Hyp	Ref	Expression
1		upgr11wlkd.j	. . 3 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
2		preq2 4213	. . . . . . 7 ⊢ (𝑌 = 𝑋 → {𝑋, 𝑌} = {𝑋, 𝑋})
3	2	eqeq2d 2620	. . . . . 6 ⊢ (𝑌 = 𝑋 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} ↔ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋}))
4	3	eqcoms 2618	. . . . 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 4138	. . . . . . 7 ⊢ {𝑋} = {𝑋, 𝑋}
7	5, 6	syl6eqr 2662	. . . . . 6 ⊢ ((((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} ∧ 𝜑) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
8	7	ex 449	. . . . 5 ⊢ (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑋} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋}))
9	4, 8	syl6bi 242	. . . 4 ⊢ (𝑋 = 𝑌 → (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋})))
10	9	com13 86	. . 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 444	1 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  {cpr 4127  ‘cfv 5804  ⟨“cs1 13149  ⟨“cs2 13437  Vtxcvtx 25673  iEdgciedg 25674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128

This theorem is referenced by:  upgr11wlkd  41314  upgr1trld  41315  upgr1pthd  41316  upgr1pthond  41317