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Theorem resdm2 5352
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 5324 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  ( A  |`  dom  `' `' A )
2 relcnv 5234 . . 3  |-  Rel  `' `' A
3 resdm 5176 . . 3  |-  ( Rel  `' `' A  ->  ( `' `' A  |`  dom  `' `' A )  =  `' `' A )
42, 3ax-mp 8 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  `' `' A
5 dmcnvcnv 5084 . . 3  |-  dom  `' `' A  =  dom  A
65reseq2i 5135 . 2  |-  ( A  |`  dom  `' `' A
)  =  ( A  |`  dom  A )
71, 4, 63eqtr3ri 2464 1  |-  ( A  |`  dom  A )  =  `' `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1652   `'ccnv 4869   dom cdm 4870    |` cres 4872   Rel wrel 4875
This theorem is referenced by:  resdmres  5353  fimacnvinrn  24039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882
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