Theorem dprdf2

Description: The function S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016)

	Ref	Expression
Hypotheses	dprdcntz.1	$$⊢\phi \to G\mathrm{dom}\mathrm{DProd}S$$
	dprdcntz.2	$$⊢\phi \to \mathrm{dom}S=I$$
Assertion	dprdf2	$$⊢\phi \to S:I⟶\mathrm{SubGrp}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		dprdcntz.1	$$⊢\phi \to G\mathrm{dom}\mathrm{DProd}S$$
2		dprdcntz.2	$$⊢\phi \to \mathrm{dom}S=I$$
3		dprdf	$$⊢G\mathrm{dom}\mathrm{DProd}S\to S:\mathrm{dom}S⟶\mathrm{SubGrp}\left(G\right)$$
4	1 3	syl	$$⊢\phi \to S:\mathrm{dom}S⟶\mathrm{SubGrp}\left(G\right)$$
5	2	feq2d	$$⊢\phi \to \left(S:\mathrm{dom}S⟶\mathrm{SubGrp}\left(G\right)\leftrightarrow S:I⟶\mathrm{SubGrp}\left(G\right)\right)$$
6	4 5	mpbid	$$⊢\phi \to S:I⟶\mathrm{SubGrp}\left(G\right)$$