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Theorem exse2 5063
Description: Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
exse2  |-  ( R  e.  V  ->  R Se  A )

Proof of Theorem exse2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2565 . . . . 5  |-  { y  e.  A  |  y R x }  =  { y  |  ( y  e.  A  /\  y R x ) }
2 vex 2804 . . . . . . . 8  |-  y  e. 
_V
3 vex 2804 . . . . . . . 8  |-  x  e. 
_V
42, 3breldm 4899 . . . . . . 7  |-  ( y R x  ->  y  e.  dom  R )
54adantl 452 . . . . . 6  |-  ( ( y  e.  A  /\  y R x )  -> 
y  e.  dom  R
)
65abssi 3261 . . . . 5  |-  { y  |  ( y  e.  A  /\  y R x ) }  C_  dom  R
71, 6eqsstri 3221 . . . 4  |-  { y  e.  A  |  y R x }  C_  dom  R
8 dmexg 4955 . . . 4  |-  ( R  e.  V  ->  dom  R  e.  _V )
9 ssexg 4176 . . . 4  |-  ( ( { y  e.  A  |  y R x }  C_  dom  R  /\  dom  R  e.  _V )  ->  { y  e.  A  |  y R x }  e.  _V )
107, 8, 9sylancr 644 . . 3  |-  ( R  e.  V  ->  { y  e.  A  |  y R x }  e.  _V )
1110ralrimivw 2640 . 2  |-  ( R  e.  V  ->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
12 df-se 4369 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
1311, 12sylibr 203 1  |-  ( R  e.  V  ->  R Se  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   class class class wbr 4039   Se wse 4366   dom cdm 4705
This theorem is referenced by:  dfac8clem  7675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-se 4369  df-cnv 4713  df-dm 4715  df-rn 4716
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