Theorem resdm 4794

Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resdm	|- ( Rel A -> ( A |` dom A ) = A )

Proof of Theorem resdm

Step	Hyp	Ref	Expression
1		ssid 3067	. 2 |- dom A C_ dom A
2		relssres 4793	. 2 |- ( ( Rel A /\ dom A C_ dom A ) -> ( A |` dom A ) = A )
3	1, 2	mpan2 419	1 |- ( Rel A -> ( A |` dom A ) = A )

Syntax hints:   -> wi 4   = wceq 1299   C_ wss 3021   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-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-dm 4487  df-res 4489

This theorem is referenced by:  resindm  4797  resdm2  4965  relresfld  5004  relcoi1  5006  funimaexg  5143  fnex  5574  dftpos2  6088  pmresg  6500  dif1en  6702