Theorem resdm2 4965

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 4937	. 2 |- ( `' `' A |` dom `' `' A ) = ( A |` dom `' `' A )
2		relcnv 4853	. . 3 |- Rel `' `' A
3		resdm 4794	. . 3 |- ( Rel `' `' A -> ( `' `' A |` dom `' `' A ) = `' `' A )
4	2, 3	ax-mp 7	. 2 |- ( `' `' A |` dom `' `' A ) = `' `' A
5		dmcnvcnv 4701	. . 3 |- dom `' `' A = dom A
6	5	reseq2i 4752	. 2 |- ( A |` dom `' `' A ) = ( A |` dom A )
7	1, 4, 6	3eqtr3ri 2129	1 |- ( A |` dom A ) = `' `' A

Syntax hints:   = wceq 1299   `'ccnv 4476   dom cdm 4477   |` cres 4479   Rel wrel 4482

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489

This theorem is referenced by:  resdmres  4966