### Nuprl Lemma : transitive-closure-map

`∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].`
`  ∀f:A ⟶ A. ((∀x,y:A.  ((R x y) `` (R (f x) (f y)))) `` (∀x,y:A.  ((TC(R) x y) `` (TC(R) (f x) (f y)))))`

Proof

Definitions occuring in Statement :  transitive-closure: `TC(R)` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` transitive-closure: `TC(R)` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` subtype_rel: `A ⊆r B` spreadn: spread3 cand: `A c∧ B` squash: `↓T` label: `...\$L... t` top: `Top` uimplies: `b supposing a` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` pi1: `fst(t)` pi2: `snd(t)` rel_path: `rel_path(A;L;x;y)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]`
Lemmas referenced :  transitive-closure_wf all_wf map_wf less_than_wf squash_wf true_wf equal_wf length-map-sq subtype_rel_list top_wf length_wf iff_weakening_equal rel_path_wf list_induction list_wf list_ind_nil_lemma map_nil_lemma list_ind_cons_lemma map_cons_lemma and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation rename sqequalHypSubstitution sqequalRule applyEquality cut introduction extract_by_obid isectElimination thin cumulativity hypothesisEquality functionExtensionality hypothesis lambdaEquality functionEquality universeEquality setElimination dependent_set_memberEquality productElimination productEquality because_Cache dependent_pairEquality independent_pairFormation imageElimination equalityTransitivity equalitySymmetry intEquality natural_numberEquality isect_memberEquality voidElimination voidEquality independent_isectElimination imageMemberEquality baseClosed independent_functionElimination dependent_functionElimination applyLambdaEquality

Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].
\mforall{}f:A  {}\mrightarrow{}  A
((\mforall{}x,y:A.    ((R  x  y)  {}\mRightarrow{}  (R  (f  x)  (f  y))))  {}\mRightarrow{}  (\mforall{}x,y:A.    ((TC(R)  x  y)  {}\mRightarrow{}  (TC(R)  (f  x)  (f  y)))))

Date html generated: 2017_04_17-AM-09_25_42
Last ObjectModification: 2017_02_27-PM-05_26_24

Theory : relations2

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