Nuprl Lemma : fun_with_inv_is_bij
∀[A,B:Type]. ∀f:A ⟶ B. ∀g:B ⟶ A. Bij(A;B;f) supposing InvFuns(A;B;f;g)
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
inv_funs: InvFuns(A;B;f;g),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
prop: ℙ,
and: P ∧ Q,
inv_funs: InvFuns(A;B;f;g),
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
biject: Bij(A;B;f),
compose: f o g,
tidentity: Id{T},
identity: Id,
exists: ∃x:A. B[x]
Lemmas referenced :
inv_funs_wf,
equal_wf,
and_wf
Rules used in proof :
universeEquality,
Error :inhabitedIsType,
Error :functionIsType,
hypothesisEquality,
isectElimination,
extract_by_obid,
Error :universeIsType,
rename,
hypothesis,
axiomEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
Error :equalityIsType1,
independent_pairFormation,
setElimination,
applyLambdaEquality,
equalityTransitivity,
dependent_set_memberEquality,
equalitySymmetry,
hyp_replacement,
Error :dependent_pairFormation_alt
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. \mforall{}g:B {}\mrightarrow{} A. Bij(A;B;f) supposing InvFuns(A;B;f;g)
Date html generated:
2019_06_20-PM-00_26_35
Last ObjectModification:
2018_10_15-PM-00_56_14
Theory : fun_1
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