Nuprl Lemma : ifun_subtype_2
∀[a,b,c:ℝ].  ((a ≤ c) 
⇒ (c ≤ b) 
⇒ ({f:[a, b] ⟶ℝ| ifun(f;[a, b])}  ⊆r {f:[c, b] ⟶ℝ| ifun(f;[c, b])} ))
Proof
Definitions occuring in Statement : 
ifun: ifun(f;I)
, 
rfun: I ⟶ℝ
, 
rccint: [l, u]
, 
rleq: x ≤ y
, 
real: ℝ
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
prop: ℙ
, 
guard: {T}
, 
ifun: ifun(f;I)
, 
real-fun: real-fun(f;a;b)
, 
top: Top
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
cand: A c∧ B
Lemmas referenced : 
rfun_subtype_2, 
ifun_wf, 
rccint_wf, 
rccint-icompact, 
rfun_wf, 
rleq_transitivity, 
rleq_wf, 
real_wf, 
member_rccint_lemma, 
left_endpoint_rccint_lemma, 
right_endpoint_rccint_lemma, 
subtype_rel_sets, 
set_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
lambdaEquality, 
setElimination, 
thin, 
rename, 
dependent_set_memberEquality, 
hypothesisEquality, 
applyEquality, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
independent_functionElimination, 
hypothesis, 
sqequalRule, 
because_Cache, 
independent_isectElimination, 
dependent_functionElimination, 
productElimination, 
setEquality, 
axiomEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
productEquality, 
independent_pairFormation
Latex:
\mforall{}[a,b,c:\mBbbR{}].
    ((a  \mleq{}  c)  {}\mRightarrow{}  (c  \mleq{}  b)  {}\mRightarrow{}  (\{f:[a,  b]  {}\mrightarrow{}\mBbbR{}|  ifun(f;[a,  b])\}    \msubseteq{}r  \{f:[c,  b]  {}\mrightarrow{}\mBbbR{}|  ifun(f;[c,  b])\}  ))
Date html generated:
2016_10_26-AM-09_48_33
Last ObjectModification:
2016_08_20-AM-10_19_17
Theory : reals
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