Theorem cnvimarndm 6080

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 6068	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5686	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6056	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5894	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2768	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1539  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   “ cima 5678

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688

This theorem is referenced by:  cnvimassrndm  6150  focnvimacdmdm  6816  cnvimainrn  7067  cnrest2  23010  mbfconstlem  25376  i1fima  25427  i1fima2  25428  i1fd  25430  i1f0rn  25431  itg1addlem5  25450  fcoinver  32102  supppreima  32180  sibfof  33637  itg2addnclem  36842  itg2addnclem2  36843  ftc1anclem6  36869  f1cof1blem  46082  f1cof1b  46083  fnfocofob  46085