### Nuprl Lemma : unordered-combination_functionality

`∀[A,B:Type].  ∀n,m:ℕ.  (A ~ B `` UnorderedCombination(n;A) ~ UnorderedCombination(m;B) supposing n = m ∈ ℤ)`

Proof

Definitions occuring in Statement :  unordered-combination: `UnorderedCombination(n;T)` equipollent: `A ~ B` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` equipollent: `A ~ B` exists: `∃x:A. B[x]` prop: `ℙ` nat: `ℕ` unordered-combination: `UnorderedCombination(n;T)` and: `P ∧ Q` cand: `A c∧ B` uiff: `uiff(P;Q)` top: `Top` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` guard: `{T}` biject: `Bij(A;B;f)` inject: `Inj(A;B;f)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` surject: `Surj(A;B;f)` compose: `f o g` bag-map: `bag-map(f;bs)`
Lemmas referenced :  equal_wf equipollent_wf nat_wf bag-map_wf bag-map-no-repeats bag-size-map bag-no-repeats_wf bag-size_wf unordered-combination_wf inject_wf subtype_base_sq int_subtype_base biject-inverse squash_wf true_wf iff_weakening_equal biject_wf bag_wf bag-map-map bag-map-trivial map-id bag-subtype-list
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction axiomEquality hypothesis thin rename sqequalHypSubstitution productElimination extract_by_obid isectElimination intEquality setElimination hypothesisEquality cumulativity universeEquality dependent_set_memberEquality functionExtensionality applyEquality independent_isectElimination independent_pairFormation sqequalRule isect_memberEquality voidElimination voidEquality productEquality lambdaEquality functionEquality dependent_functionElimination equalityTransitivity equalitySymmetry because_Cache instantiate independent_functionElimination imageElimination natural_numberEquality imageMemberEquality baseClosed dependent_pairFormation applyLambdaEquality hyp_replacement

Latex:
\mforall{}[A,B:Type].
\mforall{}n,m:\mBbbN{}.    (A  \msim{}  B  {}\mRightarrow{}  UnorderedCombination(n;A)  \msim{}  UnorderedCombination(m;B)  supposing  n  =  m)

Date html generated: 2017_10_01-AM-09_05_27
Last ObjectModification: 2017_07_26-PM-04_45_40

Theory : bags

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