Theorem exse 4382

Description: Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
exse	|- ( A e. V -> R Se A )

Proof of Theorem exse

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabexg 4186	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2579	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4379	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
4	2, 3	sylibr 134	1 |- ( A e. V -> R Se A )

Syntax hints:   -> wi 4   e. wcel 2175   A.wral 2483   {crab 2487   _Vcvv 2771   class class class wbr 4043   Se wse 4375

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-sep 4161

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rab 2492  df-v 2773  df-in 3171  df-ss 3178  df-se 4379

This theorem is referenced by: (None)