Nuprl Lemma : setmem-image
∀b:coSet{i:l}. ∀f:(x:coSet{i:l} × (x ∈ b)) ⟶ coSet{i:l}.
((∀z1,z2:x:coSet{i:l} × (x ∈ b). (seteq(fst(z1);fst(z2)) ⇒ seteq(f z1;f z2)))
⇒ (∀y:coSet{i:l}. ((y ∈ set-image(f;b)) ⇐⇒ ∃x:x:coSet{i:l} × (x ∈ b). seteq(y;f x))))
Proof
Definitions occuring in Statement :
set-image: set-image(f;b),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
pi1: fst(t),
subtype_rel: A ⊆r B,
mk-coset: mk-coset(T;f),
set-dom: set-dom(s),
set-image: set-image(f;b),
set-item: set-item(s;x),
pi2: snd(t),
top: Top,
guard: {T},
squash: ↓T,
true: True,
mk-set: f"(T),
Wsup: Wsup(a;b)
Lemmas referenced :
setmem_wf,
set-image_wf,
seteq_wf,
coSet_wf,
subtype_coSet,
coSet_subtype,
setmem-iff,
mk-coset_wf,
subtype_rel_self,
mem-mk-set_wf2,
set-dom_wf,
setmem-mk-coset,
istype-void,
seteqweaken_wf,
seteq_functionality,
seteq_weakening,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
productIsType,
because_Cache,
applyEquality,
functionIsType,
inhabitedIsType,
hypothesis_subsumption,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
dependent_pairEquality_alt,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}b:coSet\{i:l\}. \mforall{}f:(x:coSet\{i:l\} \mtimes{} (x \mmember{} b)) {}\mrightarrow{} coSet\{i:l\}.
((\mforall{}z1,z2:x:coSet\{i:l\} \mtimes{} (x \mmember{} b). (seteq(fst(z1);fst(z2)) {}\mRightarrow{} seteq(f z1;f z2)))
{}\mRightarrow{} (\mforall{}y:coSet\{i:l\}. ((y \mmember{} set-image(f;b)) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:x:coSet\{i:l\} \mtimes{} (x \mmember{} b). seteq(y;f x))))
Date html generated:
2019_10_31-AM-06_34_51
Last ObjectModification:
2018_11_10-PM-00_52_16
Theory : constructive!set!theory
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