Theorem fvmptd

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

	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	1	fveq1d	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ‘ 𝐴 ) )
6	3 2	csbied	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )
7	6 4	eqeltrd	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ∈ 𝑉 )
8		eqid	⊢ ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐷 ↦ 𝐵 )
9	8	fvmpts	⊢ ( ( 𝐴 ∈ 𝐷 ∧ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
10	3 7 9	syl2anc	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
11	5 10 6	3eqtrd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )