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Theorem resdm2 4841
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 4813 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  ( A  |`  dom  `' `' A )
2 relcnv 4733 . . 3  |-  Rel  `' `' A
3 resdm 4677 . . 3  |-  ( Rel  `' `' A  ->  ( `' `' A  |`  dom  `' `' A )  =  `' `' A )
42, 3ax-mp 7 . 2  |-  ( `' `' A  |`  dom  `' `' A )  =  `' `' A
5 dmcnvcnv 4586 . . 3  |-  dom  `' `' A  =  dom  A
65reseq2i 4637 . 2  |-  ( A  |`  dom  `' `' A
)  =  ( A  |`  dom  A )
71, 4, 63eqtr3ri 2111 1  |-  ( A  |`  dom  A )  =  `' `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1285   `'ccnv 4370   dom cdm 4371    |` cres 4373   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383
This theorem is referenced by:  resdmres  4842
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