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Theorem cnvimarndm 6038
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
StepHypRef Expression
1 imadmrn 6027 . 2 (𝐴 “ dom 𝐴) = ran 𝐴
2 df-rn 5648 . . 3 ran 𝐴 = dom 𝐴
32imaeq2i 6015 . 2 (𝐴 “ ran 𝐴) = (𝐴 “ dom 𝐴)
4 dfdm4 5855 . 2 dom 𝐴 = ran 𝐴
51, 3, 43eqtr4i 2771 1 (𝐴 “ ran 𝐴) = dom 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ccnv 5636  dom cdm 5637  ran crn 5638  cima 5640
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 5260  ax-nul 5267  ax-pr 5388
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  cnvimassrndm  6108  focnvimacdmdm  6772  cnvimainrn  7021  cnrest2  22660  mbfconstlem  25014  i1fima  25065  i1fima2  25066  i1fd  25068  i1f0rn  25069  itg1addlem5  25088  fcoinver  31578  supppreima  31659  sibfof  33004  itg2addnclem  36179  itg2addnclem2  36180  ftc1anclem6  36206  f1cof1blem  45398  f1cof1b  45399  fnfocofob  45401
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