Theorem rabssab 3179

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 2423	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpr 109	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	ss2abi 3164	. 2 |- { x | ( x e. A /\ ph ) } C_ { x | ph }
4	1, 3	eqsstri 3124	1 |- { x e. A | ph } C_ { x | ph }

Syntax hints:   /\ wa 103   e. wcel 1480   {cab 2123   {crab 2418   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-in 3072  df-ss 3079

This theorem is referenced by:  epse  4259  riotasbc  5738  genipv  7310  toponsspwpwg  12178  dmtopon  12179