Theorem cnvimarndm

Description: The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009)

	Ref	Expression
Assertion	cnvimarndm	⊢ ( ◡ 𝐴 “ ran 𝐴 ) = dom 𝐴

Proof

Step	Hyp	Ref	Expression
1		imadmrn	⊢ ( ◡ 𝐴 “ dom ◡ 𝐴 ) = ran ◡ 𝐴
2		df-rn	⊢ ran 𝐴 = dom ◡ 𝐴
3	2	imaeq2i	⊢ ( ◡ 𝐴 “ ran 𝐴 ) = ( ◡ 𝐴 “ dom ◡ 𝐴 )
4		dfdm4	⊢ dom 𝐴 = ran ◡ 𝐴
5	1 3 4	3eqtr4i	⊢ ( ◡ 𝐴 “ ran 𝐴 ) = dom 𝐴