Theorem ofc1

Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014)

	Ref	Expression
Hypotheses	ofc1.1	$$⊢\phi \to A\in V$$
	ofc1.2	$$⊢\phi \to B\in W$$
	ofc1.3	$$⊢\phi \to FFnA$$
	ofc1.4	$$⊢\left(\phi \wedge X\in A\right)\to F\left(X\right)=C$$
Assertion	ofc1	$$⊢\left(\phi \wedge X\in A\right)\to \left(\left(A\times \left\{B\right\}\right)R_{f}F\right)\left(X\right)=BRC$$

Proof

Step	Hyp	Ref	Expression
1		ofc1.1	$$⊢\phi \to A\in V$$
2		ofc1.2	$$⊢\phi \to B\in W$$
3		ofc1.3	$$⊢\phi \to FFnA$$
4		ofc1.4	$$⊢\left(\phi \wedge X\in A\right)\to F\left(X\right)=C$$
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	6 3 1 1 7 9 4	ofval	$$⊢\left(\phi \wedge X\in A\right)\to \left(\left(A\times \left\{B\right\}\right)R_{f}F\right)\left(X\right)=BRC$$