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Theorem resdm2 5300
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 5272 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  ( A  |`  dom  `' `' A )
2 relcnv 5182 . . 3  |-  Rel  `' `' A
3 resdm 5124 . . 3  |-  ( Rel  `' `' A  ->  ( `' `' A  |`  dom  `' `' A )  =  `' `' A )
42, 3ax-mp 8 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  `' `' A
5 dmcnvcnv 5032 . . 3  |-  dom  `' `' A  =  dom  A
65reseq2i 5083 . 2  |-  ( A  |`  dom  `' `' A
)  =  ( A  |`  dom  A )
71, 4, 63eqtr3ri 2416 1  |-  ( A  |`  dom  A )  =  `' `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1649   `'ccnv 4817   dom cdm 4818    |` cres 4820   Rel wrel 4823
This theorem is referenced by:  resdmres  5301  fimacnvinrn  23890
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830
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