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Theorem cnvimarndm 4903
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 4891 . 2 (𝐴 “ dom 𝐴) = ran 𝐴
2 df-rn 4550 . . 3 ran 𝐴 = dom 𝐴
32imaeq2i 4879 . 2 (𝐴 “ ran 𝐴) = (𝐴 “ dom 𝐴)
4 dfdm4 4731 . 2 dom 𝐴 = ran 𝐴
51, 3, 43eqtr4i 2170 1 (𝐴 “ ran 𝐴) = dom 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1331  ccnv 4538  dom cdm 4539  ran crn 4540  cima 4542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552
This theorem is referenced by:  cnrest2  12419
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