Nuprl Lemma : fpf-join-single-property

[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[a:A]. ∀[v:B[a]]. ∀[eq:EqDecider(A)]. ∀[b:A].
  ({(↑b ∈ dom(f)) ∧ (f ⊕ v(b) f(b) ∈ B[b])}) supposing ((↑b ∈ dom(f ⊕ v)) and (b a ∈ A)))


Definitions occuring in Statement :  fpf-single: v fpf-join: f ⊕ g fpf-ap: f(x) fpf-dom: x ∈ dom(f) fpf: a:A fp-> B[a] deq: EqDecider(T) assert: b uimplies: supposing a uall: [x:A]. B[x] guard: {T} so_apply: x[s] not: ¬A and: P ∧ Q function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  guard: {T} uall: [x:A]. B[x] member: t ∈ T so_lambda: λ2x.t[x] so_apply: x[s] all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q implies:  Q or: P ∨ Q cand: c∧ B top: Top uiff: uiff(P;Q) uimplies: supposing a not: ¬A false: False subtype_rel: A ⊆B prop: bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  bfalse: ff exists: x:A. B[x] sq_type: SQType(T) bnot: ¬bb assert: b
Lemmas referenced :  fpf-join-dom fpf-single_wf fpf-single-dom fpf-dom_wf subtype-fpf2 top_wf assert_wf fpf-join_wf not_wf equal_wf deq_wf fpf_wf assert_witness fpf-join-ap-sq bool_wf eqtt_to_assert fpf-ap_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache lambdaEquality applyEquality functionExtensionality hypothesisEquality cumulativity dependent_functionElimination instantiate hypothesis productElimination independent_functionElimination unionElimination independent_pairFormation isect_memberEquality voidElimination voidEquality independent_isectElimination lambdaFormation functionEquality universeEquality isect_memberFormation independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry equalityElimination dependent_pairFormation promote_hyp

\mforall{}[A:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[f:a:A  fp->  B[a]].  \mforall{}[a:A].  \mforall{}[v:B[a]].  \mforall{}[eq:EqDecider(A)].  \mforall{}[b:A].
    (\{(\muparrow{}b  \mmember{}  dom(f))  \mwedge{}  (f  \moplus{}  a  :  v(b)  =  f(b))\})  supposing  ((\muparrow{}b  \mmember{}  dom(f  \moplus{}  a  :  v))  and  (\mneg{}(b  =  a)))

Date html generated: 2018_05_21-PM-09_29_08
Last ObjectModification: 2018_02_09-AM-10_24_11

Theory : finite!partial!functions

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