Nuprl Lemma : equipollent-function-function
∀[A,B,C:Type]. A ⟶ B ⟶ C ~ (A × B) ⟶ C
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
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,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
surject: Surj(A;B;f),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
biject_wf,
squash_wf,
true_wf,
spread_to_pi12,
iff_weakening_equal,
pair_eta_rw
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_pairFormation,
lambdaEquality,
spreadEquality,
productElimination,
thin,
independent_pairEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
cumulativity,
productEquality,
functionEquality,
independent_pairFormation,
lambdaFormation,
sqequalRule,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
applyLambdaEquality,
rename,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[A,B,C:Type]. A {}\mrightarrow{} B {}\mrightarrow{} C \msim{} (A \mtimes{} B) {}\mrightarrow{} C
Date html generated:
2017_04_17-AM-09_31_13
Last ObjectModification:
2017_02_27-PM-05_31_31
Theory : equipollence!!cardinality!
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