Theorem resdm 5102

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 3268	. 2 |- dom A C_ dom A
2		relssres 5101	. 2 |- ( ( Rel A /\ dom A C_ dom A ) -> ( A |` dom A ) = A )
3	1, 2	mpan2 429	1 |- ( Rel A -> ( A |` dom A ) = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220   dom cdm 4774   |` cres 4776   Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-res 4786

This theorem is used by:  resindm  5105  resdm2  5278  relresfld  5317  relcoi1  5319  funimaexg  5465  fnex  5937  dftpos2  6532  pmresg  6957  dif1en  7183