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Theorem cnvimarndm 5260
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 5248 . 2  |-  ( `' A " dom  `' A )  =  ran  `' A
2 df-rn 4924 . . 3  |-  ran  A  =  dom  `' A
32imaeq2i 5236 . 2  |-  ( `' A " ran  A
)  =  ( `' A " dom  `' A )
4 dfdm4 5098 . 2  |-  dom  A  =  ran  `' A
51, 3, 43eqtr4i 2473 1  |-  ( `' A " ran  A
)  =  dom  A
Colors of variables: wff set class
Syntax hints:    = wceq 1654   `'ccnv 4912   dom cdm 4913   ran crn 4914   "cima 4916
This theorem is referenced by:  cnrest2  17388  mbfconstlem  19557  i1fima  19606  i1fima2  19607  i1fd  19609  i1f0rn  19610  itg1addlem5  19628  sibfof  24689  itg2addnclem  26298  itg2addnclem2  26299  ftc1anclem6  26327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-xp 4919  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
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