Nuprl Lemma : subtype_rel_set

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

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]` 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` subtype_rel: `A ⊆r B` so_apply: `x[s]` prop: `ℙ`
Lemmas referenced :  subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaEquality setElimination thin rename hypothesisEquality applyEquality hypothesis sqequalHypSubstitution sqequalRule setEquality because_Cache axiomEquality lemma_by_obid isectElimination isect_memberEquality equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality

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

Date html generated: 2016_05_13-PM-03_18_45
Last ObjectModification: 2015_12_26-AM-09_08_20

Theory : subtype_0

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