Theorem exse 4218

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 4031	. . 3 |- ( A e. V -> { y e. A | y R x } e. _V )
2	1	ralrimivw 2480	. 2 |- ( A e. V -> A. x e. A { y e. A | y R x } e. _V )
3		df-se 4215	. 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 1463   A.wral 2390   {crab 2394   _Vcvv 2657   class class class wbr 3895   Se wse 4211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-in 3043  df-ss 3050  df-se 4215

This theorem is referenced by: (None)