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Metamath Proof Explorer

Theorem rexsupp

Description: Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015) (Revised by AV, 27-May-2019)

Ref Expression
Assertion rexsupp ( ( 𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ ( 𝐹 supp 𝑍 ) 𝜑 ↔ ∃ 𝑥 ∈ 𝑋 ( ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ∧ 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 elsuppfn ⊢ ( ( 𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝑥 ∈ ( 𝐹 supp 𝑍 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ) )
2 1 anbi1d ⊢ ( ( 𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( ( 𝑥 ∈ ( 𝐹 supp 𝑍 ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ∧ 𝜑 ) ) )
3 anass ⊢ ( ( ( 𝑥 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ( ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ∧ 𝜑 ) ) )
4 2 3 bitrdi ⊢ ( ( 𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( ( 𝑥 ∈ ( 𝐹 supp 𝑍 ) ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ( ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ∧ 𝜑 ) ) ) )
5 4 rexbidv2 ⊢ ( ( 𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ ( 𝐹 supp 𝑍 ) 𝜑 ↔ ∃ 𝑥 ∈ 𝑋 ( ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ∧ 𝜑 ) ) )