Theorem fvmptd

Description: Deduction version of fvmpt . (Contributed by Scott Fenton, 18-Feb-2013) (Revised by Mario Carneiro, 31-Aug-2015) (Proof shortened by AV, 29-Mar-2024)

	Ref	Expression
Hypotheses	fvmptd.1	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
	fvmptd.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
	fvmptd.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
	fvmptd.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
Assertion	fvmptd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fvmptd.1	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
2		fvmptd.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
3		fvmptd.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
4		fvmptd.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
5		nfv	⊢ Ⅎ 𝑥 𝜑
6		nfcv	⊢ Ⅎ 𝑥 𝐴
7		nfcv	⊢ Ⅎ 𝑥 𝐶
8	1 2 3 4 5 6 7	fvmptdf	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )