# Metamath Proof Explorer

## Theorem isomuspgrlem2e

Description: Lemma 5 for isomuspgrlem2 . (Contributed by AV, 1-Dec-2022)

Ref Expression
Hypotheses isomushgr.v 𝑉 = ( Vtx ‘ 𝐴 )
isomushgr.w 𝑊 = ( Vtx ‘ 𝐵 )
isomushgr.e 𝐸 = ( Edg ‘ 𝐴 )
isomushgr.k 𝐾 = ( Edg ‘ 𝐵 )
isomuspgrlem2.g 𝐺 = ( 𝑥𝐸 ↦ ( 𝐹𝑥 ) )
isomuspgrlem2.a ( 𝜑𝐴 ∈ USPGraph )
isomuspgrlem2.f ( 𝜑𝐹 : 𝑉1-1-onto𝑊 )
isomuspgrlem2.i ( 𝜑 → ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) )
isomuspgrlem2.x ( 𝜑𝐹𝑋 )
isomuspgrlem2.b ( 𝜑𝐵 ∈ USPGraph )
Assertion isomuspgrlem2e ( 𝜑𝐺 : 𝐸1-1-onto𝐾 )

### Proof

Step Hyp Ref Expression
1 isomushgr.v 𝑉 = ( Vtx ‘ 𝐴 )
2 isomushgr.w 𝑊 = ( Vtx ‘ 𝐵 )
3 isomushgr.e 𝐸 = ( Edg ‘ 𝐴 )
4 isomushgr.k 𝐾 = ( Edg ‘ 𝐵 )
5 isomuspgrlem2.g 𝐺 = ( 𝑥𝐸 ↦ ( 𝐹𝑥 ) )
6 isomuspgrlem2.a ( 𝜑𝐴 ∈ USPGraph )
7 isomuspgrlem2.f ( 𝜑𝐹 : 𝑉1-1-onto𝑊 )
8 isomuspgrlem2.i ( 𝜑 → ∀ 𝑎𝑉𝑏𝑉 ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { ( 𝐹𝑎 ) , ( 𝐹𝑏 ) } ∈ 𝐾 ) )
9 isomuspgrlem2.x ( 𝜑𝐹𝑋 )
10 isomuspgrlem2.b ( 𝜑𝐵 ∈ USPGraph )
11 1 2 3 4 5 6 7 8 9 isomuspgrlem2c ( 𝜑𝐺 : 𝐸1-1𝐾 )
12 1 2 3 4 5 6 7 8 9 10 isomuspgrlem2d ( 𝜑𝐺 : 𝐸onto𝐾 )
13 df-f1o ( 𝐺 : 𝐸1-1-onto𝐾 ↔ ( 𝐺 : 𝐸1-1𝐾𝐺 : 𝐸onto𝐾 ) )
14 11 12 13 sylanbrc ( 𝜑𝐺 : 𝐸1-1-onto𝐾 )