Nuprl Lemma : mem_functionality_wrt_permr
∀s:DSet. ∀a,b:|s|. ∀as,bs:|s| List. ((a = b ∈ |s|) ⇒ (as ≡(|s|) bs) ⇒ a ∈b as = b ∈b bs)
Proof
Definitions occuring in Statement :
mem: a ∈b as,
permr: as ≡(T) bs,
list: T List,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
mem: a ∈b as,
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
prop: ℙ,
subtype_rel: A ⊆r B,
bool: 𝔹,
grp_car: |g|,
pi1: fst(t),
bor_mon: <𝔹,∨b>,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
infix_ap: x f y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
permr_wf,
set_car_wf,
list_wf,
dset_wf,
bool_wf,
bor_mon_wf,
abmonoid_subtype_iabmonoid,
infix_ap_wf,
grp_car_wf,
set_eq_wf,
subtype_rel_self,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
subtype_rel_transitivity,
abmonoid_wf,
mon_wf,
grp_sig_wf,
mem_f_wf,
mon_for_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mon_for_functionality_wrt_permr,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
equalityIsType1,
inhabitedIsType,
applyEquality,
sqequalRule,
lambdaEquality_alt,
because_Cache,
functionEquality,
instantiate,
independent_isectElimination,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}s:DSet. \mforall{}a,b:|s|. \mforall{}as,bs:|s| List. ((a = b) {}\mRightarrow{} (as \mequiv{}(|s|) bs) {}\mRightarrow{} a \mmember{}\msubb{} as = b \mmember{}\msubb{} bs)
Date html generated:
2019_10_16-PM-01_03_27
Last ObjectModification:
2018_10_08-AM-11_21_55
Theory : list_2
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