Theorem resdm2 4831

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

Step	Hyp	Ref	Expression
1		rescnvcnv 4803	. 2 |- ( `' `' A |` dom `' `' A ) = ( A |` dom `' `' A )
2		relcnv 4723	. . 3 |- Rel `' `' A
3		resdm 4667	. . 3 |- ( Rel `' `' A -> ( `' `' A |` dom `' `' A ) = `' `' A )
4	2, 3	ax-mp 7	. 2 |- ( `' `' A |` dom `' `' A ) = `' `' A
5		dmcnvcnv 4576	. . 3 |- dom `' `' A = dom A
6	5	reseq2i 4627	. 2 |- ( A |` dom `' `' A ) = ( A |` dom A )
7	1, 4, 6	3eqtr3ri 2110	1 |- ( A |` dom A ) = `' `' A

Syntax hints:   = wceq 1284   `'ccnv 4362   dom cdm 4363   |` cres 4365   Rel wrel 4368

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375

This theorem is referenced by:  resdmres  4832