Theorem elsuppfnd

Description: Deduce membership in the support of a function. (Contributed by Thierry Arnoux, 5-Oct-2025)

	Ref	Expression
Hypotheses	elsuppfnd.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
	elsuppfnd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	elsuppfnd.3	⊢ ( 𝜑 → 𝑍 ∈ 𝑊 )
	elsuppfnd.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	elsuppfnd.5	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ≠ 𝑍 )
Assertion	elsuppfnd	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐹 supp 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		elsuppfnd.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2		elsuppfnd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3		elsuppfnd.3	⊢ ( 𝜑 → 𝑍 ∈ 𝑊 )
4		elsuppfnd.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
5		elsuppfnd.5	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ≠ 𝑍 )
6		elsuppfn	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝑋 ∈ ( 𝐹 supp 𝑍 ) ↔ ( 𝑋 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑋 ) ≠ 𝑍 ) ) )
7	6	biimpar	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ ( 𝑋 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑋 ) ≠ 𝑍 ) ) → 𝑋 ∈ ( 𝐹 supp 𝑍 ) )
8	1 2 3 4 5 7	syl32anc	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐹 supp 𝑍 ) )