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 ∖ { ∅ } ) ∧ 𝐴 ≠ 𝐵 ∧ { 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 ) , ( 1st ‘ 𝐺 ) , ( Base ‘ 𝐺 ) )
4	1 2	fun2dmnop0	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴 ≠ 𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → ¬ 𝐺 ∈ ( V × V ) )
5	4	iffalsed	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴 ≠ 𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → if ( 𝐺 ∈ ( V × V ) , ( 1st ‘ 𝐺 ) , ( Base ‘ 𝐺 ) ) = ( Base ‘ 𝐺 ) )
6	3 5	syl5eq	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝐴 ≠ 𝐵 ∧ { 𝐴 , 𝐵 } ⊆ dom 𝐺 ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )