Theorem cnvimarndm 6082

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

Assertion

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

Proof of Theorem cnvimarndm

Step	Hyp	Ref	Expression
1		imadmrn 6070	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5688	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6058	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5896	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2771	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1542  ^◡ccnv 5676  dom cdm 5677  ran crn 5678   “ cima 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690

This theorem is referenced by:  cnvimassrndm  6152  focnvimacdmdm  6818  cnvimainrn  7069  cnrest2  22790  mbfconstlem  25144  i1fima  25195  i1fima2  25196  i1fd  25198  i1f0rn  25199  itg1addlem5  25218  fcoinver  31835  supppreima  31913  sibfof  33339  itg2addnclem  36539  itg2addnclem2  36540  ftc1anclem6  36566  f1cof1blem  45784  f1cof1b  45785  fnfocofob  45787