Theorem relssres 5437

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 473	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> Rel A )
2		vex 3203	. . . . . . . . 9 |- x e. _V
3		vex 3203	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 5328	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5		ssel 3597	. . . . . . . 8 |- ( dom A C_ B -> ( x e. dom A -> x e. B ) )
6	4, 5	syl5 34	. . . . . . 7 |- ( dom A C_ B -> ( <. x , y >. e. A -> x e. B ) )
7	6	ancld 576	. . . . . 6 |- ( dom A C_ B -> ( <. x , y >. e. A -> ( <. x , y >. e. A /\ x e. B ) ) )
8	3	opelres 5401	. . . . . 6 |- ( <. x , y >. e. ( A |` B ) <-> ( <. x , y >. e. A /\ x e. B ) )
9	7, 8	syl6ibr 242	. . . . 5 |- ( dom A C_ B -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
10	9	adantl 482	. . . 4 |- ( ( Rel A /\ dom A C_ B ) -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
11	1, 10	relssdv 5212	. . 3 |- ( ( Rel A /\ dom A C_ B ) -> A C_ ( A |` B ) )
12		resss 5422	. . 3 |- ( A |` B ) C_ A
13	11, 12	jctil 560	. 2 |- ( ( Rel A /\ dom A C_ B ) -> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
14		eqss 3618	. 2 |- ( ( A |` B ) = A <-> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
15	13, 14	sylibr 224	1 |- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   dom cdm 5114   |` cres 5116   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-res 5126

This theorem is referenced by:  resdm  5441  resid  5460  fnresdm  6000  f1ompt  6382  tfr2b  7492  tz7.48-2  7537  omxpenlem  8061  rankwflemb  8656  zorn2lem4  9321  relexpaddg  13793  setscom  15903  setsid  15914  dprd2da  18441  dprd2db  18442  ustssco  22018  dvres3  23677  dvres3a  23678  rlimcnp2  24693  ex-res  27298  nolt02o  31845  nosupbnd1  31860  poimirlem3  33412  relexpaddss  38010  fnresdmss  39348  limsupresuz  39935  liminfresuz  40016