Theorem exse 4314

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 4125	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2540	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4311	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
4	2, 3	sylibr 133	1 |- ( A e. V -> R Se A )

Syntax hints:   -> wi 4   e. wcel 2136   A.wral 2444   {crab 2448   _Vcvv 2726   class class class wbr 3982   Se wse 4307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-se 4311

This theorem is referenced by: (None)