Theorem ofcfval2

Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017)

	Ref	Expression
Hypotheses	ofcfval2.1	$$⊢\phi \to A\in V$$
	ofcfval2.2	$$⊢\phi \to C\in W$$
	ofcfval2.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in X$$
	ofcfval2.4	$$⊢\phi \to F=\left(x\in A⟼B\right)$$
Assertion	ofcfval2	$$⊢\phi \to F{\circ }_{\mathrm{fc}}RC=\left(x\in A⟼BRC\right)$$

Proof

Step	Hyp	Ref	Expression
1		ofcfval2.1	$$⊢\phi \to A\in V$$
2		ofcfval2.2	$$⊢\phi \to C\in W$$
3		ofcfval2.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in X$$
4		ofcfval2.4	$$⊢\phi \to F=\left(x\in A⟼B\right)$$
5	3	ralrimiva	$$⊢\phi \to \forall x\in AB\in X$$
6		eqid	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
7	6	fnmpt	$$⊢\forall x\in AB\in X\to \left(x\in A⟼B\right)FnA$$
8	5 7	syl	$$⊢\phi \to \left(x\in A⟼B\right)FnA$$
9	4	fneq1d	$$⊢\phi \to \left(FFnA\leftrightarrow \left(x\in A⟼B\right)FnA\right)$$
10	8 9	mpbird	$$⊢\phi \to FFnA$$
11	4 3	fvmpt2d	$$⊢\left(\phi \wedge x\in A\right)\to F\left(x\right)=B$$
12	10 1 2 11	ofcfval	$$⊢\phi \to F{\circ }_{\mathrm{fc}}RC=\left(x\in A⟼BRC\right)$$