Theorem upgr1wlkdlem1

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

	Ref	Expression
Hypotheses	upgr1wlkd.p	⊢ 𝑃 = ⟨“ 𝑋 𝑌 ”⟩
	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	upgr1wlkdlem1	⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } )

Proof

Step	Hyp	Ref	Expression
1		upgr1wlkd.p	⊢ 𝑃 = ⟨“ 𝑋 𝑌 ”⟩
2		upgr1wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 ”⟩
3		upgr1wlkd.x	⊢ ( 𝜑 → 𝑋 ∈ ( Vtx ‘ 𝐺 ) )
4		upgr1wlkd.y	⊢ ( 𝜑 → 𝑌 ∈ ( Vtx ‘ 𝐺 ) )
5		upgr1wlkd.j	⊢ ( 𝜑 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } )
6		preq2	⊢ ( 𝑌 = 𝑋 → { 𝑋 , 𝑌 } = { 𝑋 , 𝑋 } )
7	6	eqeq2d	⊢ ( 𝑌 = 𝑋 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑋 } ) )
8	7	eqcoms	⊢ ( 𝑋 = 𝑌 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑋 } ) )
9		simpl	⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑋 } ∧ 𝜑 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑋 } )
10		dfsn2	⊢ { 𝑋 } = { 𝑋 , 𝑋 }
11	9 10	eqtr4di	⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑋 } ∧ 𝜑 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } )
12	11	ex	⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑋 } → ( 𝜑 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } ) )
13	8 12	syl6bi	⊢ ( 𝑋 = 𝑌 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } → ( 𝜑 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } ) ) )
14	13	com13	⊢ ( 𝜑 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 , 𝑌 } → ( 𝑋 = 𝑌 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } ) ) )
15	5 14	mpd	⊢ ( 𝜑 → ( 𝑋 = 𝑌 → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } ) )
16	15	imp	⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝐽 ) = { 𝑋 } )