Nuprl Lemma : inv-rel-inject
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[finv:B ⟶ (A?)]. Inj(A;B;f) supposing inv-rel(A;B;f;finv)
Proof
Definitions occuring in Statement :
inv-rel: inv-rel(A;B;f;finv),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
unit: Unit,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
inject: Inj(A;B;f),
inv-rel: inv-rel(A;B;f;finv),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
outl: outl(x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True
Lemmas referenced :
equal_wf,
all_wf,
unit_wf2,
and_wf,
outl_wf,
assert_wf,
isl_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
because_Cache,
productEquality,
functionEquality,
unionEquality,
inlEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
independent_isectElimination,
promote_hyp,
natural_numberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[finv:B {}\mrightarrow{} (A?)]. Inj(A;B;f) supposing inv-rel(A;B;f;finv)
Date html generated:
2018_05_21-PM-06_50_17
Last ObjectModification:
2018_05_19-PM-04_41_18
Theory : general
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