### Nuprl Lemma : subtype_rel_nested_set2

`∀[A,B:Type]. ∀[P:B ⟶ ℙ]. ∀[Q:{b:B| P[b]}  ⟶ ℙ].  {b:{b:B| P[b]} | Q[b]}  ⊆r A supposing {b:B| P[b] ∧ Q[b]}  ⊆r A`

Proof

Definitions occuring in Statement :  uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` and: `P ∧ Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_apply: `x[s]` subtype_rel: `A ⊆r B` and: `P ∧ Q` prop: `ℙ`
Lemmas referenced :  subtype_rel_transitivity subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin setEquality hypothesisEquality applyEquality hypothesis because_Cache sqequalRule independent_isectElimination lambdaEquality setElimination rename dependent_set_memberEquality independent_pairFormation productEquality axiomEquality cumulativity isect_memberEquality equalityTransitivity equalitySymmetry functionEquality universeEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[P:B  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[Q:\{b:B|  P[b]\}    {}\mrightarrow{}  \mBbbP{}].
\{b:\{b:B|  P[b]\}  |  Q[b]\}    \msubseteq{}r  A  supposing  \{b:B|  P[b]  \mwedge{}  Q[b]\}    \msubseteq{}r  A

Date html generated: 2016_05_13-PM-03_18_50
Last ObjectModification: 2015_12_26-AM-09_08_19

Theory : subtype_0

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