Theorem grpinvfn

Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015)

	Ref	Expression
Hypotheses	grpinvfn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpinvfn.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion	grpinvfn	⊢ 𝑁 Fn 𝐵

Proof

Step	Hyp	Ref	Expression
1		grpinvfn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpinvfn.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
3		riotaex	⊢ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ∈ V
4		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
5		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
6	1 4 5 2	grpinvfval	⊢ 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) )
7	3 6	fnmpti	⊢ 𝑁 Fn 𝐵