Nuprl Lemma : member-mapfilter
∀[T:Type]
∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹.
∀[T':Type]
∀f:{x:T| (x ∈ L) c∧ (↑(P x))} ⟶ T'. ∀x:T'.
((x ∈ mapfilter(f;P;L)) ⇐⇒ ∃y:T. ((y ∈ L) ∧ ((↑(P y)) c∧ (x = (f y) ∈ T'))))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T
Lemmas referenced :
member-mapfilter-witness_wf
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
hypothesis
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}.
\mforall{}[T':Type]
\mforall{}f:\{x:T| (x \mmember{} L) c\mwedge{} (\muparrow{}(P x))\} {}\mrightarrow{} T'. \mforall{}x:T'.
((x \mmember{} mapfilter(f;P;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((y \mmember{} L) \mwedge{} ((\muparrow{}(P y)) c\mwedge{} (x = (f y)))))
Date html generated:
2016_05_14-AM-07_50_39
Last ObjectModification:
2015_12_26-PM-04_46_02
Theory : list_1
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