Theorem resdm 5083

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 3262	. 2 |- dom A C_ dom A
2		relssres 5082	. 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 1398   C_ wss 3214   dom cdm 4755   |` cres 4757   Rel wrel 4760

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-xp 4761  df-rel 4762  df-dm 4765  df-res 4767

This theorem is referenced by:  resindm  5086  resdm2  5259  relresfld  5298  relcoi1  5300  funimaexg  5446  fnex  5912  dftpos2  6506  pmresg  6924  dif1en  7150