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Theorem relssres 5309
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 3116 . . . . . . . . 9  |-  x  e. 
_V
3 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 5204 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
5 ssel 3498 . . . . . . . 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 5277 . . . . . 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 5093 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  B )  ->  A  C_  ( A  |`  B ) )
12 resss 5295 . . 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 3519 . 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 1379    e. wcel 1767    C_ wss 3476   <.cop 4033   dom cdm 4999    |` cres 5001   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-dm 5009  df-res 5011
This theorem is referenced by:  resdm  5313  resid  5329  fnresdm  5688  f1ompt  6041  tfr2b  7062  tz7.48-2  7104  omxpenlem  7615  rankwflemb  8207  zorn2lem4  8875  setscom  14516  setsid  14527  dprd2da  16881  dprd2db  16882  ustssco  20452  dvres3  22052  dvres3a  22053  rlimcnp2  23024  constr3pthlem1  24331  ex-res  24839  relexpadd  28536  nofulllem3  29041  nofulllem5  29043  fnresdmss  31021
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