Theorem rabssab 3272

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 2484	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpr 110	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	ss2abi 3256	. 2 |- { x | ( x e. A /\ ph ) } C_ { x | ph }
4	1, 3	eqsstri 3216	1 |- { x e. A | ph } C_ { x | ph }

Syntax hints:   /\ wa 104   e. wcel 2167   {cab 2182   {crab 2479   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-in 3163  df-ss 3170

This theorem is referenced by:  epse  4378  riotasbc  5896  genipv  7593  toponsspwpwg  14342  dmtopon  14343