Nuprl Lemma : funinv_wf2
∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. (inv(f) ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} )
Proof
Definitions occuring in Statement :
funinv: inv(f),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
and: P ∧ Q,
int_seg: {i..j-},
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
cand: A c∧ B
Lemmas referenced :
set_wf,
int_seg_wf,
inject_wf,
nat_wf,
subtype_rel_sets,
all_wf,
equal_wf,
and_wf,
funinv_wf,
surject_wf,
injection-is-surjection
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
thin,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
lambdaEquality,
isect_memberEquality,
because_Cache,
applyEquality,
productEquality,
intEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
productElimination,
dependent_functionElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} ]. (inv(f) \mmember{} \{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} )
Date html generated:
2016_05_14-AM-07_30_35
Last ObjectModification:
2015_12_26-PM-01_25_52
Theory : int_2
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