Nuprl Lemma : fset-image-compose
∀[T,A,B:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[eqb:EqDecider(B)]. ∀[f:T ⟶ A]. ∀[g:A ⟶ B]. ∀[s:fset(T)].
(g"(f"(s)) = g o f"(s) ∈ fset(B))
Proof
Definitions occuring in Statement :
fset-image: f"(s),
fset: fset(T),
deq: EqDecider(T),
compose: f o g,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
false: False,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
not: ¬A,
compose: f o g,
cand: A c∧ B,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset-extensionality,
fset-image_wf,
compose_wf,
decidable__fset-member,
fset-member_witness,
fset-member_wf,
fset_wf,
deq_wf,
squash_wf,
exists_wf,
equal_wf,
true_wf,
iff_weakening_equal,
member-fset-image-iff,
iff_transitivity,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_pairFormation,
because_Cache,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
lambdaEquality,
productEquality,
imageElimination,
dependent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
lambdaFormation,
promote_hyp,
existsFunctionality,
andLevelFunctionality
Latex:
\mforall{}[T,A,B:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[eqb:EqDecider(B)]. \mforall{}[f:T {}\mrightarrow{} A].
\mforall{}[g:A {}\mrightarrow{} B]. \mforall{}[s:fset(T)].
(g"(f"(s)) = g o f"(s))
Date html generated:
2017_04_17-AM-09_21_07
Last ObjectModification:
2017_02_27-PM-05_24_16
Theory : finite!sets
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