Theorem relssres 4922

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 108	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> Rel A )
2		vex 2729	. . . . . . . . 9 |- x e. _V
3		vex 2729	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4807	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5		ssel 3136	. . . . . . . 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 323	. . . . . 6 |- ( dom A C_ B -> ( <. x , y >. e. A -> ( <. x , y >. e. A /\ x e. B ) ) )
8	3	opelres 4889	. . . . . 6 |- ( <. x , y >. e. ( A |` B ) <-> ( <. x , y >. e. A /\ x e. B ) )
9	7, 8	syl6ibr 161	. . . . 5 |- ( dom A C_ B -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
10	9	adantl 275	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
11	1, 10	relssdv 4696	. . 3 |- ( ( Rel A /\ dom A C_ B ) -> A C_ ( A |` B ) )
12		resss 4908	. . 3 |- ( A |` B ) C_ A
13	11, 12	jctil 310	. 2 |- ( ( Rel A /\ dom A C_ B ) -> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
14		eqss 3157	. 2 |- ( ( A |` B ) = A <-> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
15	13, 14	sylibr 133	1 |- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   C_ wss 3116   <.cop 3579   dom cdm 4604   |` cres 4606   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614  df-res 4616

This theorem is referenced by:  resdm  4923  resid  4940  fnresdm  5297  f1ompt  5636  setscom  12434  setsslid  12444