Theorem fvmptdv2

Description: Alternate deduction version of fvmpt , suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017)

	Ref	Expression
Hypotheses	fvmptdv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
	fvmptdv2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 ∈ 𝑉 )
	fvmptdv2.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
Assertion	fvmptdv2	⊢ ( 𝜑 → ( 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fvmptdv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
2		fvmptdv2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 ∈ 𝑉 )
3		fvmptdv2.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
4		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
5	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
6		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
7	5 6	sylib	⊢ ( 𝜑 → ∃ 𝑥 𝑥 = 𝐴 )
8	2	elexd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 ∈ V )
9	3 8	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐶 ∈ V )
10	7 9	exlimddv	⊢ ( 𝜑 → 𝐶 ∈ V )
11	4 3 1 10	fvmptd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ‘ 𝐴 ) = 𝐶 )
12		fveq1	⊢ ( 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) = ( ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ‘ 𝐴 ) )
13	12	eqeq1d	⊢ ( 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐶 ↔ ( ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ‘ 𝐴 ) = 𝐶 ) )
14	11 13	syl5ibrcom	⊢ ( 𝜑 → ( 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) = 𝐶 ) )