Theorem cnvimarndm 6085

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 6072	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5672	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6060	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5885	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2796	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1570  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is referenced by:  cnvimassrndm  6149  focnvimacdmdm  6804  cnvimainrn  7062  cnrest2  23443  mbfconstlem  25786  i1fima  25837  i1fima2  25838  i1fd  25840  i1f0rn  25841  itg1addlem5  25859  fcoinver  32949  supppreima  33036  sibfof  34730  itg2addnclem  38342  itg2addnclem2  38343  ftc1anclem6  38369  f1cof1blem  47831  f1cof1b  47834  fnfocofob  47836