### Nuprl Lemma : funinv-unique

`∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. ∀[g:ℕn ⟶ ℕn].`
`  inv(f) = g ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)}  supposing (f o g) = (λx.x) ∈ (ℕn ⟶ ℕn)`

Proof

Definitions occuring in Statement :  funinv: `inv(f)` inject: `Inj(A;B;f)` compose: `f o g` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` prop: `ℙ` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` inject: `Inj(A;B;f)` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` implies: `P `` Q` compose: `f o g` and: `P ∧ Q` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top`
Lemmas referenced :  funinv_wf2 inject_wf int_seg_wf set_wf equal-wf-T-base compose_wf nat_wf equal_wf funinv-property int_seg_properties lelt_wf nat_properties decidable__equal_int satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_wf decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma decidable__lt intformless_wf int_formula_prop_less_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality setElimination rename dependent_set_memberEquality hypothesis natural_numberEquality because_Cache functionExtensionality applyEquality lambdaEquality sqequalRule imageMemberEquality baseClosed imageElimination functionEquality isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry dependent_functionElimination setEquality independent_functionElimination hyp_replacement Error :applyLambdaEquality,  productElimination intEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  ].  \mforall{}[g:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n].    inv(f)  =  g  supposing  (f  o  g)  =  (\mlambda{}x.x)

Date html generated: 2016_10_21-AM-09_59_48
Last ObjectModification: 2016_07_12-AM-05_14_19

Theory : int_2

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