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Theorem relssres 5248
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 3074 . . . . . . . . 9  |-  x  e. 
_V
3 vex 3074 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 5144 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
5 ssel 3451 . . . . . . . 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 5217 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( A  |`  B )  <-> 
( <. x ,  y
>.  e.  A  /\  x  e.  B ) )
97, 8syl6ibr 227 . . . . 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 5033 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  A  C_  ( A  |`  B ) )
12 resss 5235 . . 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 3472 . 2  |-  ( ( A  |`  B )  =  A  <->  ( ( A  |`  B )  C_  A  /\  A  C_  ( A  |`  B ) ) )
1513, 14sylibr 212 1  |-  ( ( Rel  A  /\  dom  A 
C_  B )  -> 
( A  |`  B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3429   <.cop 3984   dom cdm 4941    |` cres 4943   Rel wrel 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-dm 4951  df-res 4953
This theorem is referenced by:  resdm  5249  resid  5264  fnresdm  5621  f1ompt  5967  tfr2b  6958  tz7.48-2  7000  omxpenlem  7515  rankwflemb  8104  zorn2lem4  8772  setscom  14315  setsid  14326  dprd2da  16655  dprd2db  16656  ustssco  19914  dvres3  21514  dvres3a  21515  rlimcnp2  22486  constr3pthlem1  23686  ex-res  23793  relexpadd  27477  nofulllem3  27982  nofulllem5  27984
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