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Metamath Proof Explorer

Theorem fvmptd2

Description: Deduction version of fvmpt (where the definition of the mapping does not depend on the common antecedent ph ). (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	fvmptd2.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 )
	fvmptd2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
	fvmptd2.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
	fvmptd2.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
Assertion	fvmptd2	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fvmptd2.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 )
2		fvmptd2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
3		fvmptd2.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
4		fvmptd2.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
5	1	a1i	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
6	5 2 3 4	fvmptd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )