Theorem funvtxval0

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

	Ref	Expression
Hypothesis	funvtxval0.s	⊢ 𝑆 ∈ V
Assertion	funvtxval0	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝑆 ≠ ( Base ‘ ndx ) ∧ { ( Base ‘ ndx ) , 𝑆 } ⊆ dom 𝐺 ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		funvtxval0.s	⊢ 𝑆 ∈ V
2		necom	⊢ ( 𝑆 ≠ ( Base ‘ ndx ) ↔ ( Base ‘ ndx ) ≠ 𝑆 )
3		fvex	⊢ ( Base ‘ ndx ) ∈ V
4	3 1	funvtxdm2val	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ ( Base ‘ ndx ) ≠ 𝑆 ∧ { ( Base ‘ ndx ) , 𝑆 } ⊆ dom 𝐺 ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
5	2 4	syl3an2b	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 𝑆 ≠ ( Base ‘ ndx ) ∧ { ( Base ‘ ndx ) , 𝑆 } ⊆ dom 𝐺 ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )