Nuprl Lemma : subtype-fpf2
∀[A:Type]. ∀[B1,B2:A ⟶ Type].  a:A fp-> B1[a] ⊆r a:A fp-> B2[a] supposing ∀a:A. (B1[a] ⊆r B2[a])
Proof
Definitions occuring in Statement : 
fpf: a:A fp-> B[a]
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
fpf: a:A fp-> B[a]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
all: ∀x:A. B[x]
Lemmas referenced : 
subtype_rel_product, 
list_wf, 
l_member_wf, 
subtype_rel_dep_function, 
set_wf, 
all_wf, 
subtype_rel_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
introduction, 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
lambdaEquality, 
functionEquality, 
setEquality, 
applyEquality, 
setElimination, 
rename, 
because_Cache, 
independent_isectElimination, 
lambdaFormation, 
dependent_functionElimination, 
axiomEquality, 
isect_memberEquality, 
equalityTransitivity, 
equalitySymmetry, 
cumulativity, 
universeEquality
Latex:
\mforall{}[A:Type].  \mforall{}[B1,B2:A  {}\mrightarrow{}  Type].    a:A  fp->  B1[a]  \msubseteq{}r  a:A  fp->  B2[a]  supposing  \mforall{}a:A.  (B1[a]  \msubseteq{}r  B2[a])
Date html generated:
2018_05_21-PM-09_17_05
Last ObjectModification:
2018_02_09-AM-10_16_22
Theory : finite!partial!functions
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