### 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))].`
`  A = B ∈ nat-trans(C;D;F;G) supposing A = 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: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  prop: `ℙ` all: `∀x:A. B[x]` so_apply: `x[s]` so_lambda: `λ2x.t[x]` nat-trans: `nat-trans(C;D;F;G)` uimplies: `b 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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