Theorem ss2rab 2132

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 1659	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2		df-rab 1659	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
3	1, 2	sseq12i 2096	. 2 |- ({x e. A | ph} (_ {x e. A | ps} <-> {x | (x e. A /\ ph)} (_ {x | (x e. A /\ ps)})
4		ss2ab 2125	. 2 |- ({x | (x e. A /\ ph)} (_ {x | (x e. A /\ ps)} <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
5		df-ral 1656	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		imdistan 445	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
7	6	albii 1005	. . 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 958   e. wcel 962  {cab 1470  A.wral 1652  {crab 1655   (_ wss 2056

This theorem is referenced by:  ss2rabdv 2136  ss2rabi 2137  scottex 4728  ondomon 4869  uzwo3lem1 6225  uzwo3lem2 6226  occont 9167  hsupss 9316  spanss 9325  chpssat 10298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-ral 1656  df-rab 1659  df-in 2060  df-ss 2062

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