Nuprl Lemma : fpf-all_wf
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[P:x:{x:A| ↑x ∈ dom(f)} ⟶ B[x] ⟶ ℙ].
(∀x∈dom(f). v=f(x) ⇒ P[x;v] ∈ ℙ)
Proof
Definitions occuring in Statement :
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
so_apply: x[s1;s2]
Lemmas referenced :
all_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-ap_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionEquality,
because_Cache,
applyEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
universeEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
setElimination,
rename
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[P:x:\{x:A| \muparrow{}x \mmember{} dom(f)\}
{}\mrightarrow{} B[x]
{}\mrightarrow{} \mBbbP{}].
(\mforall{}x\mmember{}dom(f). v=f(x) {}\mRightarrow{} P[x;v] \mmember{} \mBbbP{})
Date html generated:
2018_05_21-PM-09_26_23
Last ObjectModification:
2018_02_09-AM-10_21_51
Theory : finite!partial!functions
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