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Metamath Proof Explorer

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 )