Nuprl Lemma : fset-constrained-image_functionality_wrt_subset
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[f:T ⟶ A]. ∀[P:A ⟶ 𝔹]. ∀[s1,s2:fset(T)].
f"(s1) s.t. P ⊆ f"(s2) s.t. P supposing s1 ⊆ s2
Proof
Definitions occuring in Statement :
fset-constrained-image: f"(s) s.t. P,
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
f-subset: xs ⊆ ys,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T}
Lemmas referenced :
fset-member_witness,
fset-constrained-image_wf,
fset-member_wf,
f-subset_wf,
fset_wf,
bool_wf,
deq_wf,
member-fset-constrained-image-iff,
equal_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionEquality,
universeEquality,
productElimination,
independent_isectElimination,
imageElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
productEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[s1,s2:fset(T)].
f"(s1) s.t. P \msubseteq{} f"(s2) s.t. P supposing s1 \msubseteq{} s2
Date html generated:
2017_04_17-AM-09_21_23
Last ObjectModification:
2017_02_27-PM-05_24_21
Theory : finite!sets
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