Theorem relssres 4648

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 102	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> Rel A )
2		vex 2560	. . . . . . . . 9 |- x e. _V
3		vex 2560	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4538	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5		ssel 2939	. . . . . . . 8 |- ( dom A C_ B -> ( x e. dom A -> x e. B ) )
6	4, 5	syl5 28	. . . . . . 7 |- ( dom A C_ B -> ( <. x , y >. e. A -> x e. B ) )
7	6	ancld 308	. . . . . 6 |- ( dom A C_ B -> ( <. x , y >. e. A -> ( <. x , y >. e. A /\ x e. B ) ) )
8	3	opelres 4617	. . . . . 6 |- ( <. x , y >. e. ( A |` B ) <-> ( <. x , y >. e. A /\ x e. B ) )
9	7, 8	syl6ibr 151	. . . . 5 |- ( dom A C_ B -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
10	9	adantl 262	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
11	1, 10	relssdv 4432	. . 3 |- ( ( Rel A /\ dom A C_ B ) -> A C_ ( A |` B ) )
12		resss 4635	. . 3 |- ( A |` B ) C_ A
13	11, 12	jctil 295	. 2 |- ( ( Rel A /\ dom A C_ B ) -> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
14		eqss 2960	. 2 |- ( ( A |` B ) = A <-> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
15	13, 14	sylibr 137	1 |- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   C_ wss 2917   <.cop 3378   dom cdm 4345   |` cres 4347   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-res 4357

This theorem is referenced by:  resdm  4649  resid  4662  fnresdm  5008  f1ompt  5320