Nuprl Lemma : equipollent-iff-inverse-funs
∀[A,B:Type]. (A ~ B ⇐⇒ ∃p:{A ⟶ B × (B ⟶ A)| InvFuns(A;B;fst(p);snd(p))})
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
inv_funs: InvFuns(A;B;f;g),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
sq_exists: ∃x:{A| B[x]},
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
squash: ↓T,
all: ∀x:A. B[x],
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
biject-iff-inverse,
inv_funs_wf,
sq_exists_wf,
pi2_wf,
pi1_wf,
sq_stable__inv_funs,
biject_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
functionEquality,
hypothesisEquality,
lambdaEquality,
hypothesis,
setElimination,
rename,
independent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality,
universeEquality,
dependent_functionElimination,
dependent_set_memberFormation,
independent_pairEquality,
dependent_pairFormation
Latex:
\mforall{}[A,B:Type]. (A \msim{} B \mLeftarrow{}{}\mRightarrow{} \mexists{}p:\{A {}\mrightarrow{} B \mtimes{} (B {}\mrightarrow{} A)| InvFuns(A;B;fst(p);snd(p))\})
Date html generated:
2016_05_14-PM-03_59_43
Last ObjectModification:
2016_01_14-PM-11_06_47
Theory : equipollence!!cardinality!
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