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Theorem dfres3 25382
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 4890 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 eleq1 2496 . . . . . . . . . 10  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  <->  <. y ,  z
>.  e.  A ) )
3 vex 2959 . . . . . . . . . . . 12  |-  z  e. 
_V
43biantru 492 . . . . . . . . . . 11  |-  ( y  e.  B  <->  ( y  e.  B  /\  z  e.  _V ) )
5 vex 2959 . . . . . . . . . . . . 13  |-  y  e. 
_V
65, 3opelrn 5101 . . . . . . . . . . . 12  |-  ( <.
y ,  z >.  e.  A  ->  z  e. 
ran  A )
76biantrud 494 . . . . . . . . . . 11  |-  ( <.
y ,  z >.  e.  A  ->  ( y  e.  B  <->  ( y  e.  B  /\  z  e.  ran  A ) ) )
84, 7syl5bbr 251 . . . . . . . . . 10  |-  ( <.
y ,  z >.  e.  A  ->  ( ( y  e.  B  /\  z  e.  _V )  <->  ( y  e.  B  /\  z  e.  ran  A ) ) )
92, 8syl6bi 220 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  A  ->  ( (
y  e.  B  /\  z  e.  _V )  <->  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
109com12 29 . . . . . . . 8  |-  ( x  e.  A  ->  (
x  =  <. y ,  z >.  ->  (
( y  e.  B  /\  z  e.  _V ) 
<->  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
1110pm5.32d 621 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  B  /\  z  e.  ran  A ) ) ) )
12112exbidv 1638 . . . . . 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 4895 . . . . . 6  |-  ( x  e.  ( B  X.  _V )  <->  E. y E. z
( x  =  <. y ,  z >.  /\  (
y  e.  B  /\  z  e.  _V )
) )
14 elxp 4895 . . . . . 6  |-  ( x  e.  ( B  X.  ran  A )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  B  /\  z  e.  ran  A ) ) )
1512, 13, 143bitr4g 280 . . . . 5  |-  ( x  e.  A  ->  (
x  e.  ( B  X.  _V )  <->  x  e.  ( B  X.  ran  A
) ) )
1615pm5.32i 619 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( B  X.  _V ) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  ran  A ) ) )
17 elin 3530 . . . 4  |-  ( x  e.  ( A  i^i  ( B  X.  ran  A
) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  ran  A ) ) )
1816, 17bitr4i 244 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  X.  _V ) )  <->  x  e.  ( A  i^i  ( B  X.  ran  A ) ) )
1918ineqri 3534 . 2  |-  ( A  i^i  ( B  X.  _V ) )  =  ( A  i^i  ( B  X.  ran  A ) )
201, 19eqtri 2456 1  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   <.cop 3817    X. cxp 4876   ran crn 4879    |` cres 4880
This theorem is referenced by:  brrestrict  25794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890
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