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Metamath Proof Explorer

Theorem fvmptd3

Description: Deduction version of fvmpt . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses fvmptd3.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 )
fvmptd3.2 ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
fvmptd3.3 ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
fvmptd3.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
Assertion fvmptd3 ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )

Proof

Step Hyp Ref Expression
1 fvmptd3.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐷 ↦ 𝐵 )
2 fvmptd3.2 ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
3 fvmptd3.3 ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
4 fvmptd3.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
5 2 1 fvmptg ⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉 ) → ( 𝐹 ‘ 𝐴 ) = 𝐶 )
6 3 4 5 syl2anc ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 )