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Theorem cnvimarndm 5034
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 5024 . 2  |-  ( `' A " dom  `' A )  =  ran  `' A
2 df-rn 4700 . . 3  |-  ran  A  =  dom  `' A
32imaeq2i 5010 . 2  |-  ( `' A " ran  A
)  =  ( `' A " dom  `' A )
4 dfdm4 4872 . 2  |-  dom  A  =  ran  `' A
51, 3, 43eqtr4i 2313 1  |-  ( `' A " ran  A
)  =  dom  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692
This theorem is referenced by:  cnrest2  17014  mbfconstlem  18984  i1fima  19033  i1fima2  19034  i1fd  19036  i1f0rn  19037  itg1addlem5  19055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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