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Theorem dfse2 5001
Description: Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dfse2  |-  ( R Se  A  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
Distinct variable groups:    x, A    x, R

Proof of Theorem dfse2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-se 4333 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
2 dfrab3 3411 . . . . 5  |-  { y  e.  A  |  y R x }  =  ( A  i^i  { y  |  y R x } )
3 vex 2740 . . . . . . 7  |-  x  e. 
_V
4 iniseg 5000 . . . . . . 7  |-  ( x  e.  _V  ->  ( `' R " { x } )  =  {
y  |  y R x } )
53, 4ax-mp 5 . . . . . 6  |-  ( `' R " { x } )  =  {
y  |  y R x }
65ineq2i 3333 . . . . 5  |-  ( A  i^i  ( `' R " { x } ) )  =  ( A  i^i  { y  |  y R x }
)
72, 6eqtr4i 2201 . . . 4  |-  { y  e.  A  |  y R x }  =  ( A  i^i  ( `' R " { x } ) )
87eleq1i 2243 . . 3  |-  ( { y  e.  A  | 
y R x }  e.  _V  <->  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
98ralbii 2483 . 2  |-  ( A. x  e.  A  {
y  e.  A  | 
y R x }  e.  _V  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
101, 9bitri 184 1  |-  ( R Se  A  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   {crab 2459   _Vcvv 2737    i^i cin 3128   {csn 3592   class class class wbr 4003   Se wse 4329   `'ccnv 4625   "cima 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-se 4333  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by:  isoselem  5820
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