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Theorem resdm2 5323
Description: A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
resdm2  |-  ( A  |`  dom  A )  =  `' `' A

Proof of Theorem resdm2
StepHypRef Expression
1 rescnvcnv 5296 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  ( A  |`  dom  `' `' A )
2 relcnv 5201 . . 3  |-  Rel  `' `' A
3 resdm 5143 . . 3  |-  ( Rel  `' `' A  ->  ( `' `' A  |`  dom  `' `' A )  =  `' `' A )
42, 3ax-mp 5 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  `' `' A
5 dmcnvcnv 5057 . . 3  |-  dom  `' `' A  =  dom  A
65reseq2i 5102 . 2  |-  ( A  |`  dom  `' `' A
)  =  ( A  |`  dom  A )
71, 4, 63eqtr3ri 2467 1  |-  ( A  |`  dom  A )  =  `' `' A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   `'ccnv 4834   dom cdm 4835    |` cres 4837   Rel wrel 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847
This theorem is referenced by:  resdmres  5324  fimacnvinrn  25903
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