Theorem elbasfv

Description: Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015)

	Ref	Expression
Hypotheses	elbasfv.s	⊢ 𝑆 = ( 𝐹 ‘ 𝑍 )
	elbasfv.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
Assertion	elbasfv	⊢ ( 𝑋 ∈ 𝐵 → 𝑍 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		elbasfv.s	⊢ 𝑆 = ( 𝐹 ‘ 𝑍 )
2		elbasfv.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
3		n0i	⊢ ( 𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅ )
4		fvprc	⊢ ( ¬ 𝑍 ∈ V → ( 𝐹 ‘ 𝑍 ) = ∅ )
5	1 4	eqtrid	⊢ ( ¬ 𝑍 ∈ V → 𝑆 = ∅ )
6	5	fveq2d	⊢ ( ¬ 𝑍 ∈ V → ( Base ‘ 𝑆 ) = ( Base ‘ ∅ ) )
7		base0	⊢ ∅ = ( Base ‘ ∅ )
8	6 2 7	3eqtr4g	⊢ ( ¬ 𝑍 ∈ V → 𝐵 = ∅ )
9	3 8	nsyl2	⊢ ( 𝑋 ∈ 𝐵 → 𝑍 ∈ V )