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Theorem dfse2 5046
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 4353 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
2 dfrab3 3444 . . . . 5  |-  { y  e.  A  |  y R x }  =  ( A  i^i  { y  |  y R x } )
3 vex 2791 . . . . . . 7  |-  x  e. 
_V
4 iniseg 5044 . . . . . . 7  |-  ( x  e.  _V  ->  ( `' R " { x } )  =  {
y  |  y R x } )
53, 4ax-mp 8 . . . . . 6  |-  ( `' R " { x } )  =  {
y  |  y R x }
65ineq2i 3367 . . . . 5  |-  ( A  i^i  ( `' R " { x } ) )  =  ( A  i^i  { y  |  y R x }
)
72, 6eqtr4i 2306 . . . 4  |-  { y  e.  A  |  y R x }  =  ( A  i^i  ( `' R " { x } ) )
87eleq1i 2346 . . 3  |-  ( { y  e.  A  | 
y R x }  e.  _V  <->  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
98ralbii 2567 . 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 240 1  |-  ( R Se  A  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151   {csn 3640   class class class wbr 4023   Se wse 4350   `'ccnv 4688   "cima 4692
This theorem is referenced by:  isoselem  5838  fnse  6232
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-se 4353  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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