Nuprl Lemma : uequiv_rel_self_functionality

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (UniformEquivRel(T;x,y.R[x;y]) `` {∀[a,a',b,b':T].  (R[a;b] `` R[a';b'] `` (R[a;a'] `⇐⇒` R[b;b']))})`

Proof

Definitions occuring in Statement :  uequiv_rel: `UniformEquivRel(T;x,y.E[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_apply: `x[s1;s2]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  guard: `{T}` uall: `∀[x:A]. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` uequiv_rel: `UniformEquivRel(T;x,y.E[x; y])` utrans: `UniformlyTrans(T;x,y.E[x; y])` usym: `UniformlySym(T;x,y.E[x; y])` urefl: `UniformlyRefl(T;x,y.E[x; y])` member: `t ∈ T` prop: `ℙ` so_apply: `x[s1;s2]` rev_implies: `P `` Q` so_lambda: `λ2x y.t[x; y]`
Lemmas referenced :  uequiv_rel_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin applyEquality hypothesisEquality Error :inhabitedIsType,  Error :universeIsType,  cut introduction extract_by_obid isectElimination lambdaEquality hypothesis Error :functionIsType,  universeEquality independent_functionElimination because_Cache

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(UniformEquivRel(T;x,y.R[x;y])
{}\mRightarrow{}  \{\mforall{}[a,a',b,b':T].    (R[a;b]  {}\mRightarrow{}  R[a';b']  {}\mRightarrow{}  (R[a;a']  \mLeftarrow{}{}\mRightarrow{}  R[b;b']))\})

Date html generated: 2019_06_20-PM-00_29_08
Last ObjectModification: 2018_09_26-AM-11_57_50

Theory : rel_1

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