Theorem exse 4457

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 4255	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2616	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4454	. 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 2203   A.wral 2520   {crab 2524   _Vcvv 2813   class class class wbr 4109   Se wse 4450

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-sep 4228

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rab 2529  df-v 2815  df-in 3217  df-ss 3224  df-se 4454

This theorem is referenced by: (None)