Theorem resdm 5441

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 3624	. 2 |- dom A C_ dom A
2		relssres 5437	. 2 |- ( ( Rel A /\ dom A C_ dom A ) -> ( A |` dom A ) = A )
3	1, 2	mpan2 707	1 |- ( Rel A -> ( A |` dom A ) = A )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   dom cdm 5114   |` cres 5116   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-res 5126

This theorem is referenced by:  resindm  5444  resdm2  5624  relresfld  5662  fnex  6481  dftpos2  7369  tfrlem11  7484  tfrlem15  7488  tfrlem16  7489  pmresg  7885  domss2  8119  axdc3lem4  9275  gruima  9624  funresdm1  29416  bnj1321  31095  funsseq  31666  nosupbnd2lem1  31861  nosupbnd2  31862  noetalem2  31864  noetalem3  31865  alrmomo2  34124  seff  38508  sblpnf  38509