Nuprl Lemma : equipollent-product-product
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[C:a:A ⟶ B[a] ⟶ Type]. x:A ⟶ y:B[x] ⟶ C[x;y] ~ p:(a:A × B[a]) ⟶ C[fst(p);snd(p)]
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
pi1: fst(t),
pi2: snd(t),
so_apply: x[s],
so_apply: x[s1;s2],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
surject: Surj(A;B;f),
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
istype-universe,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
productElimination,
thin,
sqequalRule,
applyEquality,
hypothesisEquality,
Error :productIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
Error :universeIsType,
Error :functionIsType,
independent_pairFormation,
Error :lambdaFormation_alt,
Error :equalityIsType1,
because_Cache,
Error :inhabitedIsType,
functionEquality,
productEquality,
universeEquality,
Error :functionExtensionality_alt,
applyLambdaEquality,
Error :dependent_pairEquality_alt
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:a:A {}\mrightarrow{} B[a] {}\mrightarrow{} Type].
x:A {}\mrightarrow{} y:B[x] {}\mrightarrow{} C[x;y] \msim{} p:(a:A \mtimes{} B[a]) {}\mrightarrow{} C[fst(p);snd(p)]
Date html generated:
2019_06_20-PM-02_17_47
Last ObjectModification:
2018_10_06-AM-11_24_05
Theory : equipollence!!cardinality!
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