### Nuprl Lemma : subtype_quotient

`∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ].  T ⊆r (x,y:T//E[x;y]) supposing EquivRel(T;x,y.E[x;y])`

Proof

Definitions occuring in Statement :  equiv_rel: `EquivRel(T;x,y.E[x; y])` quotient: `x,y:A//B[x; y]` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` 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` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` cand: `A c∧ B` guard: `{T}` equiv_rel: `EquivRel(T;x,y.E[x; y])` refl: `Refl(T;x,y.E[x; y])` all: `∀x:A. B[x]` squash: `↓T` true: `True`
Lemmas referenced :  equiv_rel_wf quotient_wf member_wf squash_wf true_wf and_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaEquality pointwiseFunctionality hypothesisEquality sqequalRule axiomEquality hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity applyEquality functionExtensionality isect_memberEquality because_Cache equalityTransitivity equalitySymmetry functionEquality universeEquality independent_isectElimination independent_pairFormation pertypeMemberEquality productElimination dependent_functionElimination imageElimination natural_numberEquality imageMemberEquality baseClosed hyp_replacement dependent_set_memberEquality setElimination rename setEquality

Latex:
\mforall{}[T:Type].  \mforall{}[E:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    T  \msubseteq{}r  (x,y:T//E[x;y])  supposing  EquivRel(T;x,y.E[x;y])

Date html generated: 2016_10_21-AM-09_43_42
Last ObjectModification: 2016_07_12-AM-05_04_19

Theory : quot_1

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