Nuprl Lemma : bij_iff_1_1_corr
∀[A,B:Type]. (∃f:A ⟶ B. Bij(A;B;f) ⇐⇒ 1-1-Corresp(A;B))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
one_one_corr: 1-1-Corresp(A;B),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
one_one_corr: 1-1-Corresp(A;B),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
exists_wf,
biject_wf,
inv_funs_wf,
bij_imp_exists_inv,
fun_with_inv_is_bij
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesisEquality,
lambdaEquality,
hypothesis,
Error :inhabitedIsType,
Error :universeIsType,
universeEquality,
productElimination,
dependent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination
Latex:
\mforall{}[A,B:Type]. (\mexists{}f:A {}\mrightarrow{} B. Bij(A;B;f) \mLeftarrow{}{}\mRightarrow{} 1-1-Corresp(A;B))
Date html generated:
2019_06_20-PM-00_26_37
Last ObjectModification:
2018_09_26-PM-00_11_01
Theory : fun_1
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