### Nuprl Lemma : retraction-epic

`∀[C:SmallCategory]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x y].  epic(f) supposing retraction(f)`

Proof

Definitions occuring in Statement :  cat-epic: `epic(f)` cat-retraction: `retraction(g)` 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`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` cat-epic: `epic(f)` member: `t ∈ T` cat-retraction: `retraction(g)` exists: `∃x:A. B[x]` cat-inverse: `fg=1` prop: `ℙ` true: `True` squash: `↓T` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` and: `P ∧ Q` guard: `{T}` iff: `P `⇐⇒` Q` implies: `P `` Q`
Lemmas referenced :  equal_wf cat-arrow_wf cat-comp_wf cat-retraction_wf cat-ob_wf small-category_wf squash_wf true_wf cat-comp-assoc cat-comp-ident iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin rename hypothesis extract_by_obid isectElimination applyEquality hypothesisEquality sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry applyLambdaEquality natural_numberEquality lambdaEquality imageElimination universeEquality dependent_functionElimination imageMemberEquality baseClosed independent_isectElimination independent_functionElimination

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[x,y:cat-ob(C)].  \mforall{}[f:cat-arrow(C)  x  y].    epic(f)  supposing  retraction(f)

Date html generated: 2020_05_20-AM-07_50_31
Last ObjectModification: 2017_07_28-AM-09_19_10

Theory : small!categories

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