### Nuprl Lemma : nat-trans-equal2

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

Proof

Definitions occuring in Statement :  nat-trans: `nat-trans(C;D;F;G)` functor-ob: `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 :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` nat-trans: `nat-trans(C;D;F;G)` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  equal_wf squash_wf true_wf cat-arrow_wf functor-ob_wf nat-trans-equation cat-comp_wf functor-arrow_wf cat-ob_wf iff_weakening_equal all_wf nat-trans_wf cat-functor_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation applyEquality thin lambdaEquality sqequalHypSubstitution imageElimination extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry universeEquality because_Cache functionExtensionality natural_numberEquality sqequalRule imageMemberEquality baseClosed independent_isectElimination productElimination independent_functionElimination dependent_set_memberEquality functionEquality setElimination rename isect_memberEquality axiomEquality

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

Date html generated: 2020_05_20-AM-07_51_26
Last ObjectModification: 2017_07_28-AM-09_19_17

Theory : small!categories

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