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Theorem relssres 5133
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 457 . . . 4  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  Rel  A )
2 vex 3064 . . . . . . . . 9  |-  x  e. 
_V
3 vex 3064 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 5029 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
5 ssel 3438 . . . . . . . 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 553 . . . . . 6  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  A  /\  x  e.  B ) ) )
83opelres 5101 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( A  |`  B )  <-> 
( <. x ,  y
>.  e.  A  /\  x  e.  B ) )
97, 8syl6ibr 229 . . . . 5  |-  ( dom 
A  C_  B  ->  (
<. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  ( A  |`  B )
) )
109adantl 466 . . . 4  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  ( A  |`  B )
) )
111, 10relssdv 4918 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  A  C_  ( A  |`  B ) )
12 resss 5119 . . 3  |-  ( A  |`  B )  C_  A
1311, 12jctil 537 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
14 eqss 3459 . 2  |-  ( ( A  |`  B )  =  A  <->  ( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
1513, 14sylibr 214 1  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( A  |`  B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    C_ wss 3416   <.cop 3980   dom cdm 4825    |` cres 4827   Rel wrel 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-dm 4835  df-res 4837
This theorem is referenced by:  resdm  5137  resid  5153  fnresdm  5673  f1ompt  6033  tfr2b  7101  tz7.48-2  7146  omxpenlem  7658  rankwflemb  8245  zorn2lem4  8913  relexpaddg  13037  setscom  14875  setsid  14886  dprd2da  17413  dprd2db  17414  ustssco  21011  dvres3  22611  dvres3a  22612  rlimcnp2  23624  constr3pthlem1  25084  ex-res  25592  nofulllem3  30177  nofulllem5  30179  relexpaddss  35710  fnresdmss  36831
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