Nuprl Lemma : equipollent-union-function
∀[A,B,C:Type]. (A + B) ⟶ C ~ A ⟶ C × (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],
union: left + right,
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),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
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,
pi1_wf,
pi2_wf,
squash_wf,
true_wf,
eta_conv,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_pairFormation,
lambdaEquality,
independent_pairEquality,
applyEquality,
hypothesisEquality,
inlEquality,
inrEquality,
functionEquality,
unionEquality,
independent_pairFormation,
lambdaFormation,
sqequalRule,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
universeEquality,
applyLambdaEquality,
productElimination,
functionExtensionality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[A,B,C:Type]. (A + B) {}\mrightarrow{} C \msim{} A {}\mrightarrow{} C \mtimes{} (B {}\mrightarrow{} C)
Date html generated:
2019_06_20-PM-02_16_57
Last ObjectModification:
2018_08_21-PM-01_55_46
Theory : equipollence!!cardinality!
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