Theorem ofc2

Description: Right operation by a constant. (Contributed by NM, 7-Oct-2014)

	Ref	Expression
Hypotheses	ofc2.1	$$⊢\phi \to A\in V$$
	ofc2.2	$$⊢\phi \to B\in W$$
	ofc2.3	$$⊢\phi \to FFnA$$
	ofc2.4	$$⊢\left(\phi \wedge X\in A\right)\to F\left(X\right)=C$$
Assertion	ofc2	$$⊢\left(\phi \wedge X\in A\right)\to \left(F{R}_f\left(A\times \left\{B\right\}\right)\right)\left(X\right)=CRB$$

Proof

Step	Hyp	Ref	Expression
1		ofc2.1	$$⊢\phi \to A\in V$$
2		ofc2.2	$$⊢\phi \to B\in W$$
3		ofc2.3	$$⊢\phi \to FFnA$$
4		ofc2.4	$$⊢\left(\phi \wedge X\in A\right)\to F\left(X\right)=C$$
5		fnconstg	$$⊢B\in W\to \left(A\times \left\{B\right\}\right)FnA$$
6	2 5	syl	$$⊢\phi \to \left(A\times \left\{B\right\}\right)FnA$$
7		inidm	$$⊢A\cap A=A$$
8		fvconst2g	$$⊢\left(B\in W\wedge X\in A\right)\to \left(A\times \left\{B\right\}\right)\left(X\right)=B$$
9	2 8	sylan	$$⊢\left(\phi \wedge X\in A\right)\to \left(A\times \left\{B\right\}\right)\left(X\right)=B$$
10	3 6 1 1 7 4 9	ofval	$$⊢\left(\phi \wedge X\in A\right)\to \left(F{R}_f\left(A\times \left\{B\right\}\right)\right)\left(X\right)=CRB$$