Theorem fvmpt2bd

Description: Value of a function given by the maps-to notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypothesis	fvmpt2bd.1	$$⊢\phi \to F=\left(x\in A⟼B\right)$$
Assertion	fvmpt2bd	$$⊢\left(\phi \wedge x\in A\wedge B\in C\right)\to F\left(x\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		fvmpt2bd.1	$$⊢\phi \to F=\left(x\in A⟼B\right)$$
2	1	fveq1d	$$⊢\phi \to F\left(x\right)=\left(x\in A⟼B\right)\left(x\right)$$
3	2	3ad2ant1	$$⊢\left(\phi \wedge x\in A\wedge B\in C\right)\to F\left(x\right)=\left(x\in A⟼B\right)\left(x\right)$$
4		eqid	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
5	4	fvmpt2	$$⊢\left(x\in A\wedge B\in C\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
6	5	3adant1	$$⊢\left(\phi \wedge x\in A\wedge B\in C\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
7	3 6	eqtrd	$$⊢\left(\phi \wedge x\in A\wedge B\in C\right)\to F\left(x\right)=B$$