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Theorem dfres3 24116
Description: Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfres3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A ) )

Proof of Theorem dfres3
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-res 4701 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 eleq1 2343 . . . . . . . . . 10  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  <->  <. y ,  z
>.  e.  A ) )
3 vex 2791 . . . . . . . . . . . 12  |-  z  e. 
_V
43biantru 491 . . . . . . . . . . 11  |-  ( y  e.  B  <->  ( y  e.  B  /\  z  e.  _V ) )
5 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
65, 3opelrn 4910 . . . . . . . . . . . 12  |-  ( <.
y ,  z >.  e.  A  ->  z  e. 
ran  A )
76biantrud 493 . . . . . . . . . . 11  |-  ( <.
y ,  z >.  e.  A  ->  ( y  e.  B  <->  ( y  e.  B  /\  z  e.  ran  A ) ) )
84, 7syl5bbr 250 . . . . . . . . . 10  |-  ( <.
y ,  z >.  e.  A  ->  ( ( y  e.  B  /\  z  e.  _V )  <->  ( y  e.  B  /\  z  e.  ran  A ) ) )
92, 8syl6bi 219 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  ->  ( (
y  e.  B  /\  z  e.  _V )  <->  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
109com12 27 . . . . . . . 8  |-  ( x  e.  A  ->  (
x  =  <. y ,  z >.  ->  (
( y  e.  B  /\  z  e.  _V ) 
<->  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
1110pm5.32d 620 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
12112exbidv 1614 . . . . . 6  |-  ( x  e.  A  ->  ( E. y E. z ( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
)  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  ran  A ) ) ) )
13 elxp 4706 . . . . . 6  |-  ( x  e.  ( B  X.  _V )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
) )
14 elxp 4706 . . . . . 6  |-  ( x  e.  ( B  X.  ran  A )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  B  /\  z  e.  ran  A ) ) )
1512, 13, 143bitr4g 279 . . . . 5  |-  ( x  e.  A  ->  (
x  e.  ( B  X.  _V )  <->  x  e.  ( B  X.  ran  A
) ) )
1615pm5.32i 618 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( B  X.  _V ) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  ran  A ) ) )
17 elin 3358 . . . 4  |-  ( x  e.  ( A  i^i  ( B  X.  ran  A
) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  ran  A ) ) )
1816, 17bitr4i 243 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  X.  _V ) )  <->  x  e.  ( A  i^i  ( B  X.  ran  A ) ) )
1918ineqri 3362 . 2  |-  ( A  i^i  ( B  X.  _V ) )  =  ( A  i^i  ( B  X.  ran  A ) )
201, 19eqtri 2303 1  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   <.cop 3643    X. cxp 4687   ran crn 4690    |` cres 4691
This theorem is referenced by:  brrestrict  24487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701
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