Theorem exse 5078

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 4812	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2967	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 5074	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
4	2, 3	sylibr 224	1 |- ( A e. V -> R Se A )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   Se wse 5071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-se 5074

This theorem is referenced by:  wemoiso  7153  wemoiso2  7154  oiiso  8442  hartogslem1  8447  oemapwe  8591  cantnffval2  8592  om2uzoi  12754  uzsinds  12786  bpolylem  14779