Nuprl Lemma : itersetfun_functionality
∀G:Set{i:l} ⟶ Set{i:l}
((∀a,b:Set{i:l}. ((a ⊆ b) ⇒ (G[a] ⊆ G[b])))
⇒ (∀b,a:Set{i:l}. (seteq(a;b) ⇒ seteq(itersetfun(x.G[x];a);itersetfun(x.G[x];b)))))
Proof
Definitions occuring in Statement :
itersetfun: itersetfun(s.G[s];a),
setsubset: (a ⊆ b),
seteq: seteq(s1;s2),
Set: Set{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
prop: ℙ,
cand: A c∧ B,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setsubset_wf,
all_wf,
seteq_wf,
itersetfun_functionality_subset,
Set_wf,
itersetfun_wf,
seteq-iff-setsubset
Rules used in proof :
functionEquality,
cumulativity,
instantiate,
because_Cache,
independent_pairFormation,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}G:Set\{i:l\} {}\mrightarrow{} Set\{i:l\}
((\mforall{}a,b:Set\{i:l\}. ((a \msubseteq{} b) {}\mRightarrow{} (G[a] \msubseteq{} G[b])))
{}\mRightarrow{} (\mforall{}b,a:Set\{i:l\}. (seteq(a;b) {}\mRightarrow{} seteq(itersetfun(x.G[x];a);itersetfun(x.G[x];b)))))
Date html generated:
2018_05_23-AM-08_10_11
Last ObjectModification:
2018_05_22-PM-11_28_45
Theory : constructive!set!theory
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