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Theorem cnvimarndm 5449
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 5439 . 2 (𝐴 “ dom 𝐴) = ran 𝐴
2 df-rn 5090 . . 3 ran 𝐴 = dom 𝐴
32imaeq2i 5427 . 2 (𝐴 “ ran 𝐴) = (𝐴 “ dom 𝐴)
4 dfdm4 5281 . 2 dom 𝐴 = ran 𝐴
51, 3, 43eqtr4i 2658 1 (𝐴 “ ran 𝐴) = dom 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by:  cnrest2  20995  mbfconstlem  23297  i1fima  23346  i1fima2  23347  i1fd  23349  i1f0rn  23350  itg1addlem5  23368  fcoinver  29252  sibfof  30175  itg2addnclem  33079  itg2addnclem2  33080  ftc1anclem6  33108
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