Nuprl Lemma : nat-trans-equal

[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[A:nat-trans(C;D;F;G)]. ∀[B:A:cat-ob(C) ⟶ (cat-arrow(D) (functor-ob(F) A) 
                                                                                        (functor-ob(G) A))].
  B ∈ nat-trans(C;D;F;G) supposing B ∈ (A:cat-ob(C) ⟶ (cat-arrow(D) (functor-ob(F) A) (functor-ob(G) A)))


Proof




Definitions occuring in Statement :  nat-trans: nat-trans(C;D;F;G) functor-ob: functor-ob(F) cat-functor: Functor(C1;C2) cat-arrow: cat-arrow(C) cat-ob: cat-ob(C) small-category: SmallCategory uimplies: supposing a uall: [x:A]. B[x] apply: a function: x:A ⟶ B[x] equal: t ∈ T
Definitions unfolded in proof :  prop: all: x:A. B[x] so_apply: x[s] so_lambda: λ2x.t[x] nat-trans: nat-trans(C;D;F;G) uimplies: supposing a member: t ∈ T uall: [x:A]. B[x]
Lemmas referenced :  nat-trans_wf functor-arrow_wf cat-comp_wf functor-ob_wf equal_wf cat-arrow_wf cat-ob_wf all_wf
Rules used in proof :  equalitySymmetry equalityTransitivity axiomEquality isect_memberEquality functionEquality because_Cache applyEquality lambdaEquality sqequalRule hypothesisEquality isectElimination lemma_by_obid hypothesis dependent_set_memberEquality rename thin setElimination sqequalHypSubstitution cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[C,D:SmallCategory].  \mforall{}[F,G:Functor(C;D)].  \mforall{}[A:nat-trans(C;D;F;G)].
\mforall{}[B:A:cat-ob(C)  {}\mrightarrow{}  (cat-arrow(D)  (functor-ob(F)  A)  (functor-ob(G)  A))].
    A  =  B  supposing  A  =  B



Date html generated: 2016_05_18-AM-11_52_32
Last ObjectModification: 2015_12_28-PM-02_25_09

Theory : small!categories


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