### Nuprl Lemma : fpf-join-list-dom2

`∀[A:Type]. ∀eq:EqDecider(A). ∀L:a:A fp-> Top List. ∀x:A.  (↑x ∈ dom(⊕(L)) `⇐⇒` (∃f∈L. ↑x ∈ dom(f)))`

Proof

Definitions occuring in Statement :  fpf-join-list: `⊕(L)` fpf-dom: `x ∈ dom(f)` fpf: `a:A fp-> B[a]` l_exists: `(∃x∈L. P[x])` list: `T List` deq: `EqDecider(T)` assert: `↑b` uall: `∀[x:A]. B[x]` top: `Top` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` implies: `P `` Q` fpf-join-list: `⊕(L)` top: `Top` fpf-empty: `⊗` fpf-dom: `x ∈ dom(f)` pi1: `fst(t)` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` and: `P ∧ Q` false: `False` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` or: `P ∨ Q`
Lemmas referenced :  list_induction fpf_wf top_wf all_wf iff_wf assert_wf fpf-dom_wf fpf-join-list_wf l_exists_wf l_member_wf list_wf deq_wf reduce_nil_lemma deq_member_nil_lemma false_wf l_exists_nil l_exists_wf_nil l_exists_cons cons_wf or_wf reduce_cons_lemma fpf-join-list-dom fpf-join-dom fpf-join_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin lemma_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality hypothesis cumulativity because_Cache setElimination rename setEquality independent_functionElimination dependent_functionElimination universeEquality isect_memberEquality voidElimination voidEquality introduction independent_pairFormation productElimination independent_pairEquality addLevel allFunctionality impliesFunctionality applyEquality orFunctionality levelHypothesis promote_hyp

Latex:
\mforall{}[A:Type].  \mforall{}eq:EqDecider(A).  \mforall{}L:a:A  fp->  Top  List.  \mforall{}x:A.    (\muparrow{}x  \mmember{}  dom(\moplus{}(L))  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}f\mmember{}L.  \muparrow{}x  \mmember{}  dom(f)))

Date html generated: 2018_05_21-PM-09_22_44
Last ObjectModification: 2018_02_09-AM-10_18_52

Theory : finite!partial!functions

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