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Metamath Proof Explorer

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 biimtrdi ⊢ ( 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 } )