Nuprl Lemma : fpf-rename-dom2
∀[A,C:Type]. ∀[eqa:EqDecider(A)]. ∀[eqc:EqDecider(C)]. ∀[eqc':Top]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> Top]. ∀[a:A].
{↑r a ∈ dom(rename(r;f)) supposing ↑a ∈ dom(f)}
Proof
Definitions occuring in Statement :
fpf-rename: rename(r;f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
guard: {T},
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
fpf-rename: rename(r;f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
pi1: fst(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x]
Lemmas referenced :
assert-deq-member,
map_wf,
assert_witness,
deq-member_wf,
assert_wf,
fpf-dom_wf,
fpf_wf,
top_wf,
deq_wf,
member_map,
and_wf,
l_member_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
applyEquality,
independent_functionElimination,
lambdaEquality,
functionEquality,
universeEquality,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[A,C:Type]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[eqc:EqDecider(C)]. \mforall{}[eqc':Top]. \mforall{}[r:A {}\mrightarrow{} C]. \mforall{}[f:a:A fp-> Top].
\mforall{}[a:A].
\{\muparrow{}r a \mmember{} dom(rename(r;f)) supposing \muparrow{}a \mmember{} dom(f)\}
Date html generated:
2018_05_21-PM-09_26_50
Last ObjectModification:
2018_02_09-AM-10_22_11
Theory : finite!partial!functions
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