Theorem exse 4253

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 4066	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2504	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4250	. 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 1480   A.wral 2414   {crab 2418   _Vcvv 2681   class class class wbr 3924   Se wse 4246

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-in 3072  df-ss 3079  df-se 4250

This theorem is referenced by: (None)