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Theorem dfopif 3981
Description: Rewrite df-op 3823 using  if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dfopif  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )

Proof of Theorem dfopif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-op 3823 . 2  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
2 df-3an 938 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
32abbii 2548 . 2  |-  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  { x  |  (
( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }
4 iftrue 3745 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  { { A } ,  { A ,  B } } )
5 ibar 491 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( ( A  e. 
_V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
65abbi2dv 2551 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  {
x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) } )
74, 6eqtr2d 2469 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) ) )
8 pm2.21 102 . . . . . . 7  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  e. 
_V  /\  B  e.  _V )  ->  x  e.  (/) ) )
98adantrd 455 . . . . . 6  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } )  ->  x  e.  (/) ) )
109abssdv 3417 . . . . 5  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  C_  (/) )
11 ss0 3658 . . . . 5  |-  ( { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  C_  (/)  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
1210, 11syl 16 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
13 iffalse 3746 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  (/) )
1412, 13eqtr4d 2471 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) ) )
157, 14pm2.61i 158 . 2  |-  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  if ( ( A  e. 
_V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
161, 3, 153eqtri 2460 1  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ifcif 3739   {csn 3814   {cpr 3815   <.cop 3817
This theorem is referenced by:  dfopg  3982  opeq1  3984  opeq2  3985  nfop  4000  opprc  4005  opex  4427
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-op 3823
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