Nuprl Lemma : cat-inverse-unique

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


Definitions occuring in Statement :  cat-inverse: fg=1 cat-arrow: cat-arrow(C) cat-ob: cat-ob(C) small-category: SmallCategory uimplies: supposing a uall: [x:A]. B[x] exists: x:A. B[x] apply: a equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] uimplies: supposing a exists: x:A. B[x] member: t ∈ T so_lambda: λ2x.t[x] so_apply: x[s] prop:
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

\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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