Theorem rabssab 3281

Description: A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
rabssab	|- { x e. A | ph } C_ { x | ph }

Proof of Theorem rabssab

Step	Hyp	Ref	Expression
1		df-rab 2493	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpr 110	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	ss2abi 3265	. 2 |- { x | ( x e. A /\ ph ) } C_ { x | ph }
4	1, 3	eqsstri 3225	1 |- { x e. A | ph } C_ { x | ph }

Syntax hints:   /\ wa 104   e. wcel 2176   {cab 2191   {crab 2488   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-in 3172  df-ss 3179

This theorem is referenced by:  epse  4389  riotasbc  5915  genipv  7622  toponsspwpwg  14494  dmtopon  14495