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Theorem relssres 4750
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.
StepHypRef Expression
1 simpl 107 . . . 4  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  Rel  A )
2 vex 2622 . . . . . . . . 9  |-  x  e. 
_V
3 vex 2622 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 4639 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
5 ssel 3019 . . . . . . . 8  |-  ( dom 
A  C_  B  ->  ( x  e.  dom  A  ->  x  e.  B ) )
64, 5syl5 32 . . . . . . 7  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  x  e.  B ) )
76ancld 318 . . . . . 6  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  A  /\  x  e.  B ) ) )
83opelres 4718 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( A  |`  B )  <-> 
( <. x ,  y
>.  e.  A  /\  x  e.  B ) )
97, 8syl6ibr 160 . . . . 5  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  ( A  |`  B )
) )
109adantl 271 . . . 4  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  ( A  |`  B )
) )
111, 10relssdv 4530 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  A  C_  ( A  |`  B ) )
12 resss 4737 . . 3  |-  ( A  |`  B )  C_  A
1311, 12jctil 305 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
14 eqss 3040 . 2  |-  ( ( A  |`  B )  =  A  <->  ( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
1513, 14sylibr 132 1  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( A  |`  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2999   <.cop 3449   dom cdm 4438    |` cres 4440   Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-dm 4448  df-res 4450
This theorem is referenced by:  resdm  4751  resid  4768  fnresdm  5123  f1ompt  5450  setscom  11529  setsidn  11539
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