### Nuprl Lemma : cat-inverse-unique

`∀[C:SmallCategory]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x y]. ∀[g2,g1:cat-arrow(C) y x].`
`  (g1 = g2 ∈ (cat-arrow(C) y x)) supposing ((∃h:cat-arrow(C) y x. hf=1) and fg2=1 and fg1=1)`

Proof

Definitions occuring in Statement :  cat-inverse: `fg=1` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` apply: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` exists: `∃x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ`
Lemmas referenced :  left-right-inverse-unique exists_wf cat-arrow_wf cat-inverse_wf cat-ob_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination hypothesisEquality independent_isectElimination hypothesis because_Cache applyEquality sqequalRule lambdaEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[x,y:cat-ob(C)].  \mforall{}[f:cat-arrow(C)  x  y].  \mforall{}[g2,g1:cat-arrow(C)  y  x].
(g1  =  g2)  supposing  ((\mexists{}h:cat-arrow(C)  y  x.  hf=1)  and  fg2=1  and  fg1=1)

Date html generated: 2020_05_20-AM-07_50_09
Last ObjectModification: 2017_01_08-PM-01_00_35

Theory : small!categories

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