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Theorem dfse2 5237
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 4542 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
2 dfrab3 3617 . . . . 5  |-  { y  e.  A  |  y R x }  =  ( A  i^i  { y  |  y R x } )
3 vex 2959 . . . . . . 7  |-  x  e. 
_V
4 iniseg 5235 . . . . . . 7  |-  ( x  e.  _V  ->  ( `' R " { x } )  =  {
y  |  y R x } )
53, 4ax-mp 8 . . . . . 6  |-  ( `' R " { x } )  =  {
y  |  y R x }
65ineq2i 3539 . . . . 5  |-  ( A  i^i  ( `' R " { x } ) )  =  ( A  i^i  { y  |  y R x }
)
72, 6eqtr4i 2459 . . . 4  |-  { y  e.  A  |  y R x }  =  ( A  i^i  ( `' R " { x } ) )
87eleq1i 2499 . . 3  |-  ( { y  e.  A  | 
y R x }  e.  _V  <->  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
98ralbii 2729 . 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 241 1  |-  ( R Se  A  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   {crab 2709   _Vcvv 2956    i^i cin 3319   {csn 3814   class class class wbr 4212   Se wse 4539   `'ccnv 4877   "cima 4881
This theorem is referenced by:  isoselem  6061  fnse  6463
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-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  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-se 4542  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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