Theorem exse 4321

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 4132	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2544	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4318	. 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 2141   A.wral 2448   {crab 2452   _Vcvv 2730   class class class wbr 3989   Se wse 4314

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-se 4318

This theorem is referenced by: (None)