Nuprl Lemma : function_functionality_wrt_equipollent
∀[A:Type]. ∀[B,C:A ⟶ Type]. ((∀a:A. B[a] ~ C[a]) ⇒ a:A ⟶ B[a] ~ a:A ⟶ C[a])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
pi1: fst(t),
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
biject_wf,
all_wf,
equipollent_wf,
exists_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
hypothesis,
promote_hyp,
thin,
productElimination,
dependent_pairFormation,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
functionEquality,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
applyLambdaEquality,
because_Cache,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B,C:A {}\mrightarrow{} Type]. ((\mforall{}a:A. B[a] \msim{} C[a]) {}\mRightarrow{} a:A {}\mrightarrow{} B[a] \msim{} a:A {}\mrightarrow{} C[a])
Date html generated:
2017_04_17-AM-09_31_06
Last ObjectModification:
2017_02_27-PM-05_31_25
Theory : equipollence!!cardinality!
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