Theorem resdm 4930

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 3167	. 2 |- dom A C_ dom A
2		relssres 4929	. 2 |- ( ( Rel A /\ dom A C_ dom A ) -> ( A |` dom A ) = A )
3	1, 2	mpan2 423	1 |- ( Rel A -> ( A |` dom A ) = A )

Syntax hints:   -> wi 4   = wceq 1348   C_ wss 3121   dom cdm 4611   |` cres 4613   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621  df-res 4623

This theorem is referenced by:  resindm  4933  resdm2  5101  relresfld  5140  relcoi1  5142  funimaexg  5282  fnex  5718  dftpos2  6240  pmresg  6654  dif1en  6857