Theorem dprdf2

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

	Ref	Expression
Hypotheses	dprdcntz.1	⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
	dprdcntz.2	⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
Assertion	dprdf2	⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		dprdcntz.1	⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
2		dprdcntz.2	⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
3		dprdf	⊢ ( 𝐺 dom DProd 𝑆 → 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) )
4	1 3	syl	⊢ ( 𝜑 → 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) )
5	2	feq2d	⊢ ( 𝜑 → ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ↔ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )