### Nuprl Lemma : funinv-property

`∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. ∀[x:ℕn].  (((f (inv(f) x)) = x ∈ ℤ) ∧ ((inv(f) (f x)) = x ∈ ℕn))`

Proof

Definitions occuring in Statement :  funinv: `inv(f)` inject: `Inj(A;B;f)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` and: `P ∧ Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` and: `P ∧ Q` nat: `ℕ` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` all: `∀x:A. B[x]` uimplies: `b supposing a` squash: `↓T` cand: `A c∧ B` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` inject: `Inj(A;B;f)` lelt: `i ≤ j < k` ge: `i ≥ j ` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` sq_stable: `SqStable(P)`
Lemmas referenced :  funinv_wf2 int_seg_wf set_wf inject_wf nat_wf equal_wf squash_wf injection-is-surjection surject_wf true_wf iff_weakening_equal int_seg_properties nat_properties decidable__le lelt_wf satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma sq_stable__and sq_stable__equal funinv_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality setElimination rename sqequalRule productElimination independent_pairEquality axiomEquality hypothesis natural_numberEquality isect_memberEquality because_Cache functionEquality lambdaEquality functionExtensionality applyEquality intEquality dependent_set_memberEquality setEquality dependent_functionElimination independent_isectElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination equalityTransitivity equalitySymmetry universeEquality independent_functionElimination independent_pairFormation unionElimination voidElimination dependent_pairFormation int_eqEquality voidEquality computeAll lambdaFormation

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

Date html generated: 2017_04_14-AM-09_19_17
Last ObjectModification: 2017_02_27-PM-03_55_22

Theory : int_2

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