Theorem grporndm

Description: A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008) (New usage is discouraged.)

	Ref	Expression
Assertion	grporndm	⊢ ( 𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ran 𝐺 = ran 𝐺
2	1	grpofo	⊢ ( 𝐺 ∈ GrpOp → 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 )
3		fof	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → 𝐺 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 )
4	3	fdmd	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → dom 𝐺 = ( ran 𝐺 × ran 𝐺 ) )
5	4	dmeqd	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → dom dom 𝐺 = dom ( ran 𝐺 × ran 𝐺 ) )
6		dmxpid	⊢ dom ( ran 𝐺 × ran 𝐺 ) = ran 𝐺
7	5 6	eqtr2di	⊢ ( 𝐺 : ( ran 𝐺 × ran 𝐺 ) –onto→ ran 𝐺 → ran 𝐺 = dom dom 𝐺 )
8	2 7	syl	⊢ ( 𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺 )