Theorem ss2rab 2113

Description: Restricted abstraction classes in a subclass relationship.

Assertion

Ref	Expression
ss2rab	|- ({x e. A | ph} (_ {x e. A | ps} <-> A.x e. A (ph -> ps))

Proof of Theorem ss2rab

Step	Hyp	Ref	Expression
1		df-rab 1644	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2		df-rab 1644	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
3	1, 2	sseq12i 2077	. 2 |- ({x e. A | ph} (_ {x e. A | ps} <-> {x | (x e. A /\ ph)} (_ {x | (x e. A /\ ps)})
4		ss2ab 2106	. 2 |- ({x | (x e. A /\ ph)} (_ {x | (x e. A /\ ps)} <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
5		df-ral 1641	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		imdistan 442	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
7	6	albii 996	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
8	5, 7	bitr2 174	. 2 |- (A.x((x e. A /\ ph) -> (x e. A /\ ps)) <-> A.x e. A (ph -> ps))
9	3, 4, 8	3bitr 177	1 |- ({x e. A | ph} (_ {x e. A | ps} <-> A.x e. A (ph -> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  {cab 1456  A.wral 1637  {crab 1640   (_ wss 2037

This theorem is referenced by:  ss2rabdv 2117  ss2rabi 2118  scottex 4688  ondomon 4828  uzwo3lem1 6164  uzwo3lem2 6165  occont 9076  hsupss 9224  spanss 9233  chpssat 10198

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rab 1644  df-in 2041  df-ss 2043

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