Theorem relssres 4980

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 2763	. . . . . . . . 9 |- x e. _V
3		vex 2763	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4865	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5		ssel 3173	. . . . . . . 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 4947	. . . . . 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 4751	. . 3 |- ( ( Rel A /\ dom A C_ B ) -> A C_ ( A |` B ) )
12		resss 4966	. . 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 3194	. 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 1364   e. wcel 2164   C_ wss 3153   <.cop 3621   dom cdm 4659   |` cres 4661   Rel wrel 4664

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-dm 4669  df-res 4671

This theorem is referenced by:  resdm  4981  resid  4999  fnresdm  5363  f1ompt  5709  setscom  12658  setsslid  12669