MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resdm2 Structured version   Visualization version   Unicode version

Theorem resdm2 5332
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 5305 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  ( A  |`  dom  `' `' A )
2 relcnv 5213 . . 3  |-  Rel  `' `' A
3 resdm 5152 . . 3  |-  ( Rel  `' `' A  ->  ( `' `' A  |`  dom  `' `' A )  =  `' `' A )
42, 3ax-mp 5 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  `' `' A
5 dmcnvcnv 5063 . . 3  |-  dom  `' `' A  =  dom  A
65reseq2i 5108 . 2  |-  ( A  |`  dom  `' `' A
)  =  ( A  |`  dom  A )
71, 4, 63eqtr3ri 2502 1  |-  ( A  |`  dom  A )  =  `' `' A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   `'ccnv 4838   dom cdm 4839    |` cres 4841   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851
This theorem is referenced by:  resdmres  5333  fimacnvinrn  28311
  Copyright terms: Public domain W3C validator