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Theorem cnvimarndm 5192
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 5182 . 2  |-  ( `' A " dom  `' A )  =  ran  `' A
2 df-rn 4856 . . 3  |-  ran  A  =  dom  `' A
32imaeq2i 5168 . 2  |-  ( `' A " ran  A
)  =  ( `' A " dom  `' A )
4 dfdm4 5030 . 2  |-  dom  A  =  ran  `' A
51, 3, 43eqtr4i 2442 1  |-  ( `' A " ran  A
)  =  dom  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848
This theorem is referenced by:  cnrest2  17312  mbfconstlem  19482  i1fima  19531  i1fima2  19532  i1fd  19534  i1f0rn  19535  itg1addlem5  19553  sibfof  24615  itg2addnclem  26163  itg2addnclem2  26164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
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