Nuprl Lemma : product_functionality_wrt_equipollent_dependent
∀[A,B:Type]. ∀[C:A ⟶ Type]. ∀[D:B ⟶ Type].
∀f:A ⟶ B. (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,
function: x:A ⟶ B[x],
product: 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,
so_apply: x[s],
prop: ℙ,
pi1: fst(t),
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
pi2: snd(t),
guard: {T},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent_wf,
biject_wf,
istype-universe,
equal_wf,
subtype_rel-equal,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
hypothesis,
promote_hyp,
thin,
productElimination,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
Error :inhabitedIsType,
instantiate,
universeEquality,
because_Cache,
functionExtensionality,
rename,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
Error :dependent_pairEquality_alt,
Error :productIsType,
independent_pairFormation,
productEquality,
applyLambdaEquality,
independent_isectElimination,
Error :dependent_set_memberEquality_alt,
setElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityElimination,
Error :equalityIstype
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} Type]. \mforall{}[D:B {}\mrightarrow{} Type].
\mforall{}f:A {}\mrightarrow{} B. (Bij(A;B;f) {}\mRightarrow{} (\mforall{}a:A. C[a] \msim{} D[f a]) {}\mRightarrow{} a:A \mtimes{} C[a] \msim{} b:B \mtimes{} D[b])
Date html generated:
2019_06_20-PM-02_16_49
Last ObjectModification:
2018_11_23-PM-02_55_47
Theory : equipollence!!cardinality!
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