MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvimarndm Structured version   Visualization version   GIF version

Theorem cnvimarndm 5937
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 5926 . 2 (𝐴 “ dom 𝐴) = ran 𝐴
2 df-rn 5553 . . 3 ran 𝐴 = dom 𝐴
32imaeq2i 5914 . 2 (𝐴 “ ran 𝐴) = (𝐴 “ dom 𝐴)
4 dfdm4 5751 . 2 dom 𝐴 = ran 𝐴
51, 3, 43eqtr4i 2857 1 (𝐴 “ ran 𝐴) = dom 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  ccnv 5541  dom cdm 5542  ran crn 5543  cima 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  cnrest2  21897  mbfconstlem  24237  i1fima  24288  i1fima2  24289  i1fd  24291  i1f0rn  24292  itg1addlem5  24310  fcoinver  30371  sibfof  31658  itg2addnclem  35056  itg2addnclem2  35057  ftc1anclem6  35083
  Copyright terms: Public domain W3C validator