Theorem resdm 4982

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 3200	. 2 |- dom A C_ dom A
2		relssres 4981	. 2 |- ( ( Rel A /\ dom A C_ dom A ) -> ( A |` dom A ) = A )
3	1, 2	mpan2 425	1 |- ( Rel A -> ( A |` dom A ) = A )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3154   dom cdm 4660   |` cres 4662   Rel wrel 4665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-dm 4670  df-res 4672

This theorem is referenced by:  resindm  4985  resdm2  5157  relresfld  5196  relcoi1  5198  funimaexg  5339  fnex  5781  dftpos2  6316  pmresg  6732  dif1en  6937