Theorem rabssab 3315

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 2519	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpr 110	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	ss2abi 3299	. 2 |- { x | ( x e. A /\ ph ) } C_ { x | ph }
4	1, 3	eqsstri 3259	1 |- { x e. A | ph } C_ { x | ph }

Syntax hints:   /\ wa 104   e. wcel 2202   {cab 2217   {crab 2514   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-in 3206  df-ss 3213

This theorem is referenced by:  epse  4439  riotasbc  5987  genipv  7728  toponsspwpwg  14745  dmtopon  14746