Nuprl Lemma : perm_morph_wf
∀S,T:Type. ∀s2t:S ⟶ T. ∀t2s:T ⟶ S. (InvFuns(S;T;s2t;t2s) ⇒ (∀p:Perm(S). (perm_morph(S;T;s2t;t2s;p) ∈ Perm(T))))
Proof
Definitions occuring in Statement :
perm_morph: perm_morph(S;T;s2t;t2s;p),
perm: Perm(T),
inv_funs: InvFuns(A;B;f;g),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
perm_morph: perm_morph(S;T;s2t;t2s;p),
uall: ∀[x:A]. B[x],
perm: Perm(T),
prop: ℙ,
inv_funs: InvFuns(A;B;f;g),
and: P ∧ Q,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mk_perm_wf_a,
compose_wf,
perm_f_wf,
perm_b_wf,
perm_wf,
inv_funs_wf,
istype-universe,
perm_properties,
equal_wf,
squash_wf,
true_wf,
comp_assoc,
subtype_rel_self,
comp_id_l,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
because_Cache,
setElimination,
rename,
hypothesis,
independent_functionElimination,
universeIsType,
functionIsType,
inhabitedIsType,
universeEquality,
productElimination,
independent_pairFormation,
functionEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination
Latex:
\mforall{}S,T:Type. \mforall{}s2t:S {}\mrightarrow{} T. \mforall{}t2s:T {}\mrightarrow{} S.
(InvFuns(S;T;s2t;t2s) {}\mRightarrow{} (\mforall{}p:Perm(S). (perm\_morph(S;T;s2t;t2s;p) \mmember{} Perm(T))))
Date html generated:
2019_10_16-PM-01_00_51
Last ObjectModification:
2018_10_08-AM-10_59_26
Theory : perms_2
Home
Index