Metamath Proof Explorer


Theorem funvtxdm2val

Description: The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)

Ref Expression
Hypotheses funvtxdm2val.a 𝐴 ∈ V
funvtxdm2val.b 𝐵 ∈ V
Assertion funvtxdm2val ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 funvtxdm2val.a 𝐴 ∈ V
2 funvtxdm2val.b 𝐵 ∈ V
3 vtxval ( Vtx ‘ 𝐺 ) = if ( 𝐺 ∈ ( V × V ) , ( 1st𝐺 ) , ( Base ‘ 𝐺 ) )
4 1 2 fun2dmnop0 ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → ¬ 𝐺 ∈ ( V × V ) )
5 4 iffalsed ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → if ( 𝐺 ∈ ( V × V ) , ( 1st𝐺 ) , ( Base ‘ 𝐺 ) ) = ( Base ‘ 𝐺 ) )
6 3 5 syl5eq ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )