Theorem cnvimarndm 6054

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 6041	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5649	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6029	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5859	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2762	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1540  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  cnvimassrndm  6125  focnvimacdmdm  6784  cnvimainrn  7039  cnrest2  23173  mbfconstlem  25528  i1fima  25579  i1fima2  25580  i1fd  25582  i1f0rn  25583  itg1addlem5  25601  fcoinver  32533  supppreima  32614  sibfof  34331  itg2addnclem  37665  itg2addnclem2  37666  ftc1anclem6  37692  f1cof1blem  47075  f1cof1b  47078  fnfocofob  47080