Nuprl Lemma : fpf-join-dom-decl
∀f,g:x:Id fp-> Type. ∀x:Id. (↑x ∈ dom(f ⊕ g) ⇐⇒ (↑x ∈ dom(f)) ∨ (↑x ∈ dom(g)))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
id-deq: IdDeq,
Id: Id,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
rev_implies: P ⇐ Q,
or: P ∨ Q
Lemmas referenced :
Id_wf,
fpf_wf,
or_wf,
assert_wf,
fpf-dom_wf,
id-deq_wf,
subtype-fpf2,
top_wf,
fpf-join-dom,
fpf-join_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
universeEquality,
independent_pairFormation,
hypothesisEquality,
applyEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}f,g:x:Id fp-> Type. \mforall{}x:Id. (\muparrow{}x \mmember{} dom(f \moplus{} g) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}x \mmember{} dom(f)) \mvee{} (\muparrow{}x \mmember{} dom(g)))
Date html generated:
2018_05_21-PM-09_29_50
Last ObjectModification:
2018_02_09-AM-10_24_29
Theory : finite!partial!functions
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