Theorem caofid0r

Description: Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014)

	Ref	Expression
Hypotheses	caofref.1	$$⊢\phi \to A\in V$$
	caofref.2	$$⊢\phi \to F:A⟶S$$
	caofid0.3	$$⊢\phi \to B\in W$$
	caofid0r.5	$$⊢\left(\phi \wedge x\in S\right)\to xRB=x$$
Assertion	caofid0r	$$⊢\phi \to F{R}_f\left(A\times \left\{B\right\}\right)=F$$

Proof

Step	Hyp	Ref	Expression
1		caofref.1	$$⊢\phi \to A\in V$$
2		caofref.2	$$⊢\phi \to F:A⟶S$$
3		caofid0.3	$$⊢\phi \to B\in W$$
4		caofid0r.5	$$⊢\left(\phi \wedge x\in S\right)\to xRB=x$$
5	2	ffnd	$$⊢\phi \to FFnA$$
6		fnconstg	$$⊢B\in W\to \left(A\times \left\{B\right\}\right)FnA$$
7	3 6	syl	$$⊢\phi \to \left(A\times \left\{B\right\}\right)FnA$$
8		eqidd	$$⊢\left(\phi \wedge w\in A\right)\to F\left(w\right)=F\left(w\right)$$
9		fvconst2g	$$⊢\left(B\in W\wedge w\in A\right)\to \left(A\times \left\{B\right\}\right)\left(w\right)=B$$
10	3 9	sylan	$$⊢\left(\phi \wedge w\in A\right)\to \left(A\times \left\{B\right\}\right)\left(w\right)=B$$
11	4	ralrimiva	$$⊢\phi \to \forall x\in SxRB=x$$
12	2	ffvelcdmda	$$⊢\left(\phi \wedge w\in A\right)\to F\left(w\right)\in S$$
13		oveq1	$$⊢x=F\left(w\right)\to xRB=F\left(w\right)RB$$
14		id	$$⊢x=F\left(w\right)\to x=F\left(w\right)$$
15	13 14	eqeq12d	$$⊢x=F\left(w\right)\to \left(xRB=x\leftrightarrow F\left(w\right)RB=F\left(w\right)\right)$$
16	15	rspccva	$$⊢\left(\forall x\in SxRB=x\wedge F\left(w\right)\in S\right)\to F\left(w\right)RB=F\left(w\right)$$
17	11 12 16	syl2an2r	$$⊢\left(\phi \wedge w\in A\right)\to F\left(w\right)RB=F\left(w\right)$$
18	1 5 7 5 8 10 17	offveq	$$⊢\phi \to F{R}_f\left(A\times \left\{B\right\}\right)=F$$