Theorem cnvimarndm 6040

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 6027	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5633	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6015	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5842	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2767	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1541  ^◡ccnv 5621  dom cdm 5622  ran crn 5623   “ cima 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635

This theorem is referenced by:  cnvimassrndm  6108  focnvimacdmdm  6756  cnvimainrn  7010  cnrest2  23228  mbfconstlem  25582  i1fima  25633  i1fima2  25634  i1fd  25636  i1f0rn  25637  itg1addlem5  25655  fcoinver  32628  supppreima  32719  sibfof  34446  itg2addnclem  37811  itg2addnclem2  37812  ftc1anclem6  37838  f1cof1blem  47262  f1cof1b  47265  fnfocofob  47267