Theorem rabssab 3235

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 2457	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpr 109	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	ss2abi 3219	. 2 |- { x | ( x e. A /\ ph ) } C_ { x | ph }
4	1, 3	eqsstri 3179	1 |- { x e. A | ph } C_ { x | ph }

Syntax hints:   /\ wa 103   e. wcel 2141   {cab 2156   {crab 2452   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-in 3127  df-ss 3134

This theorem is referenced by:  epse  4327  riotasbc  5824  genipv  7471  toponsspwpwg  12814  dmtopon  12815