Nuprl Lemma : isect_functionality_wrt_equipollent_dependent
∀[A,B:Type]. ∀[C:A ⟶ Type]. ∀[D:B ⟶ Type].
∀f:A ⟶ B. (A ⇒ Bij(A;B;f) ⇒ (∀[a:A]. C[a] ~ D[f a]) ⇒ ⋂a:A. C[a] ~ ⋂b:B. D[b])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
pi2: snd(t),
pi1: fst(t),
and: P ∧ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
biject: Bij(A;B;f),
uimplies: b supposing a,
top: Top,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
label: ...$L... t
Lemmas referenced :
biject_wf,
uall_wf,
equipollent_wf,
biject-inverse,
subtype_rel_self,
subtype_rel_wf,
member_wf,
squash_wf,
true_wf,
pair-eta,
isect_subtype_rel_trivial,
top_wf,
exists_wf,
subtype_rel-equal,
equal_wf,
iff_weakening_equal,
equal_functionality_wrt_subtype_rel2,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
rename,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
isectEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
lambdaEquality,
functionEquality,
universeEquality,
isect_memberEquality,
productElimination,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
productEquality,
independent_isectElimination,
independent_pairEquality,
voidElimination,
voidEquality,
dependent_pairEquality,
instantiate
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} Type]. \mforall{}[D:B {}\mrightarrow{} Type].
\mforall{}f:A {}\mrightarrow{} B. (A {}\mRightarrow{} Bij(A;B;f) {}\mRightarrow{} (\mforall{}[a:A]. C[a] \msim{} D[f a]) {}\mRightarrow{} \mcap{}a:A. C[a] \msim{} \mcap{}b:B. D[b])
Date html generated:
2017_04_17-AM-09_31_02
Last ObjectModification:
2017_02_27-PM-05_32_58
Theory : equipollence!!cardinality!
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