Nuprl Lemma : map_equal3

`∀[T,T':Type]. ∀[a:T List+]. ∀[f,g:T ⟶ T'].`
`  map(f;a) = map(g;a) ∈ T' List+ supposing ∀x:T. ((x ∈ a) `` ((f x) = (g x) ∈ T'))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` listp: `A List+` map: `map(f;as)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` listp: `A List+` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` top: `Top` or: `P ∨ Q` ge: `i ≥ j ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` true: `True` not: `¬A` false: `False` cons: `[a / b]` guard: `{T}` nat: `ℕ` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalTransitivity computationStep sqequalReflexivity functionIsType universeIsType hypothesisEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesis applyEquality lambdaEquality functionEquality inhabitedIsType because_Cache universeEquality isect_memberFormation_alt isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality voidElimination voidEquality dependent_functionElimination unionElimination productElimination independent_functionElimination natural_numberEquality promote_hyp hypothesis_subsumption lambdaFormation addEquality independent_pairFormation independent_isectElimination intEquality minusEquality

Latex:
\mforall{}[T,T':Type].  \mforall{}[a:T  List\msupplus{}].  \mforall{}[f,g:T  {}\mrightarrow{}  T'].
map(f;a)  =  map(g;a)  supposing  \mforall{}x:T.  ((x  \mmember{}  a)  {}\mRightarrow{}  ((f  x)  =  (g  x)))

Date html generated: 2019_10_15-AM-10_53_23
Last ObjectModification: 2018_09_27-AM-10_02_48

Theory : list!

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