Theorem exse 4291

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 4103	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2528	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4288	. 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 2125   A.wral 2432   {crab 2436   _Vcvv 2709   class class class wbr 3961   Se wse 4284

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136  ax-sep 4078

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rab 2441  df-v 2711  df-in 3104  df-ss 3111  df-se 4288

This theorem is referenced by: (None)