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

Theorem cnvimarndm 5943
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 5932 . 2 (𝐴 “ dom 𝐴) = ran 𝐴
2 df-rn 5559 . . 3 ran 𝐴 = dom 𝐴
32imaeq2i 5920 . 2 (𝐴 “ ran 𝐴) = (𝐴 “ dom 𝐴)
4 dfdm4 5757 . 2 dom 𝐴 = ran 𝐴
51, 3, 43eqtr4i 2851 1 (𝐴 “ ran 𝐴) = dom 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  cnrest2  21822  mbfconstlem  24155  i1fima  24206  i1fima2  24207  i1fd  24209  i1f0rn  24210  itg1addlem5  24228  fcoinver  30285  sibfof  31497  itg2addnclem  34824  itg2addnclem2  34825  ftc1anclem6  34853
  Copyright terms: Public domain W3C validator