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Theorem dfres3 28762
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 5011 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 eleq1 2539 . . . . . . . . . 10  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  <->  <. y ,  z
>.  e.  A ) )
3 vex 3116 . . . . . . . . . . . 12  |-  z  e. 
_V
43biantru 505 . . . . . . . . . . 11  |-  ( y  e.  B  <->  ( y  e.  B  /\  z  e.  _V ) )
5 vex 3116 . . . . . . . . . . . . 13  |-  y  e. 
_V
65, 3opelrn 5232 . . . . . . . . . . . 12  |-  ( <.
y ,  z >.  e.  A  ->  z  e. 
ran  A )
76biantrud 507 . . . . . . . . . . 11  |-  ( <.
y ,  z >.  e.  A  ->  ( y  e.  B  <->  ( y  e.  B  /\  z  e.  ran  A ) ) )
84, 7syl5bbr 259 . . . . . . . . . 10  |-  ( <.
y ,  z >.  e.  A  ->  ( ( y  e.  B  /\  z  e.  _V )  <->  ( y  e.  B  /\  z  e.  ran  A ) ) )
92, 8syl6bi 228 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  ->  ( (
y  e.  B  /\  z  e.  _V )  <->  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
109com12 31 . . . . . . . 8  |-  ( x  e.  A  ->  (
x  =  <. y ,  z >.  ->  (
( y  e.  B  /\  z  e.  _V ) 
<->  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
1110pm5.32d 639 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
12112exbidv 1692 . . . . . 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 5016 . . . . . 6  |-  ( x  e.  ( B  X.  _V )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
) )
14 elxp 5016 . . . . . 6  |-  ( x  e.  ( B  X.  ran  A )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  B  /\  z  e.  ran  A ) ) )
1512, 13, 143bitr4g 288 . . . . 5  |-  ( x  e.  A  ->  (
x  e.  ( B  X.  _V )  <->  x  e.  ( B  X.  ran  A
) ) )
1615pm5.32i 637 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( B  X.  _V ) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  ran  A ) ) )
17 elin 3687 . . . 4  |-  ( x  e.  ( A  i^i  ( B  X.  ran  A
) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  ran  A ) ) )
1816, 17bitr4i 252 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  X.  _V ) )  <->  x  e.  ( A  i^i  ( B  X.  ran  A ) ) )
1918ineqri 3692 . 2  |-  ( A  i^i  ( B  X.  _V ) )  =  ( A  i^i  ( B  X.  ran  A ) )
201, 19eqtri 2496 1  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    i^i cin 3475   <.cop 4033    X. cxp 4997   ran crn 5000    |` cres 5001
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-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  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-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011
This theorem is referenced by:  brrestrict  29173
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