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Metamath Proof Explorer

Theorem fmptdf

Description: A version of fmptd using bound-variable hypothesis instead of a distinct variable condition for ph . (Contributed by Glauco Siliprandi, 29-Jun-2017)

	Ref	Expression
Hypotheses	fmptdf.1	⊢ Ⅎ 𝑥 𝜑
	fmptdf.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
	fmptdf.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	fmptdf	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fmptdf.1	⊢ Ⅎ 𝑥 𝜑
2		fmptdf.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
3		fmptdf.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
4	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶 ) )
5	1 4	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
6	3	fmpt	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹 : 𝐴 ⟶ 𝐶 )
7	5 6	sylib	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )