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

Theorem cnvimarndm 5116
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 5106 . 2  |-  ( `' A " dom  `' A )  =  ran  `' A
2 df-rn 4782 . . 3  |-  ran  A  =  dom  `' A
32imaeq2i 5092 . 2  |-  ( `' A " ran  A
)  =  ( `' A " dom  `' A )
4 dfdm4 4954 . 2  |-  dom  A  =  ran  `' A
51, 3, 43eqtr4i 2388 1  |-  ( `' A " ran  A
)  =  dom  A
Colors of variables: wff set class
Syntax hints:    = wceq 1642   `'ccnv 4770   dom cdm 4771   ran crn 4772   "cima 4774
This theorem is referenced by:  cnrest2  17120  mbfconstlem  19088  i1fima  19137  i1fima2  19138  i1fd  19140  i1f0rn  19141  itg1addlem5  19159  itg2addnclem  25492  itg2addnclem2  25493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784
  Copyright terms: Public domain W3C validator