Theorem relssres 4957

Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
relssres	|- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )

Proof of Theorem relssres

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> Rel A )
2		vex 2752	. . . . . . . . 9 |- x e. _V
3		vex 2752	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4842	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5		ssel 3161	. . . . . . . 8 |- ( dom A C_ B -> ( x e. dom A -> x e. B ) )
6	4, 5	syl5 32	. . . . . . 7 |- ( dom A C_ B -> ( <. x , y >. e. A -> x e. B ) )
7	6	ancld 325	. . . . . 6 |- ( dom A C_ B -> ( <. x , y >. e. A -> ( <. x , y >. e. A /\ x e. B ) ) )
8	3	opelres 4924	. . . . . 6 |- ( <. x , y >. e. ( A |` B ) <-> ( <. x , y >. e. A /\ x e. B ) )
9	7, 8	imbitrrdi 162	. . . . 5 |- ( dom A C_ B -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
10	9	adantl 277	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
11	1, 10	relssdv 4730	. . 3 |- ( ( Rel A /\ dom A C_ B ) -> A C_ ( A |` B ) )
12		resss 4943	. . 3 |- ( A |` B ) C_ A
13	11, 12	jctil 312	. 2 |- ( ( Rel A /\ dom A C_ B ) -> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
14		eqss 3182	. 2 |- ( ( A |` B ) = A <-> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
15	13, 14	sylibr 134	1 |- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   C_ wss 3141   <.cop 3607   dom cdm 4638   |` cres 4640   Rel wrel 4643

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-dm 4648  df-res 4650

This theorem is referenced by:  resdm  4958  resid  4976  fnresdm  5337  f1ompt  5680  setscom  12515  setsslid  12526