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Theorem relssres 4945
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 109 . . . 4  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  Rel  A )
2 vex 2740 . . . . . . . . 9  |-  x  e. 
_V
3 vex 2740 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 4830 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
5 ssel 3149 . . . . . . . 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 325 . . . . . 6  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  A  /\  x  e.  B ) ) )
83opelres 4912 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( A  |`  B )  <-> 
( <. x ,  y
>.  e.  A  /\  x  e.  B ) )
97, 8imbitrrdi 162 . . . . 5  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  ( A  |`  B )
) )
109adantl 277 . . . 4  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  ( A  |`  B )
) )
111, 10relssdv 4718 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  A  C_  ( A  |`  B ) )
12 resss 4931 . . 3  |-  ( A  |`  B )  C_  A
1311, 12jctil 312 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
14 eqss 3170 . 2  |-  ( ( A  |`  B )  =  A  <->  ( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
1513, 14sylibr 134 1  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( A  |`  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    C_ wss 3129   <.cop 3595   dom cdm 4626    |` cres 4628   Rel wrel 4631
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-dm 4636  df-res 4638
This theorem is referenced by:  resdm  4946  resid  4964  fnresdm  5325  f1ompt  5667  setscom  12501  setsslid  12512
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