Nuprl Lemma : antecedent-function_functionality_wrt_iff
∀es:EO
∀[P,Q,P',Q':E ⟶ ℙ].
∀f:{e:E| P e} ⟶ {e:E| Q e} . ((∀e:E. (P e ⇐⇒ P' e)) ⇒ (∀e:E. (Q e ⇐⇒ Q' e)) ⇒ (Q ⟵─f── P ⇐⇒ Q' ⟵─f── P'))
Proof
Definitions occuring in Statement :
antecedent-function: Q ⟵─f── P,
es-E: E,
event_ordering: EO,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
antecedent-function: Q ⟵─f── P,
cand: A c∧ B,
guard: {T}
Latex:
\mforall{}es:EO
\mforall{}[P,Q,P',Q':E {}\mrightarrow{} \mBbbP{}].
\mforall{}f:\{e:E| P e\} {}\mrightarrow{} \{e:E| Q e\}
((\mforall{}e:E. (P e \mLeftarrow{}{}\mRightarrow{} P' e)) {}\mRightarrow{} (\mforall{}e:E. (Q e \mLeftarrow{}{}\mRightarrow{} Q' e)) {}\mRightarrow{} (Q \mleftarrow{}{}{}f{}{} P \mLeftarrow{}{}\mRightarrow{} Q' \mleftarrow{}{}{}f{}{} P'))
Date html generated:
2016_05_16-AM-10_27_44
Last ObjectModification:
2015_12_28-PM-09_19_06
Theory : new!event-ordering
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