ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvimarndm Unicode version

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  12414
  Copyright terms: Public domain W3C validator