Nuprl Lemma : antecedent-surjection_functionality_wrt_iff
∀es:EO
∀[P1,P2,Q1,Q2:E ⟶ ℙ].
∀f:{e:E| P1 e} ⟶ {e:E| Q1 e}
((∀e:E. (P1 e ⇐⇒ P2 e)) ⇒ (∀e:E. (Q1 e ⇐⇒ Q2 e)) ⇒ (Q1 ←⟵ f── P1 ⇐⇒ Q2 ←⟵ f── P2))
Proof
Definitions occuring in Statement :
antecedent-surjection: 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,
antecedent-surjection: Q ←⟵ f── P,
antecedent-function: Q ⟵─f── P,
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T}
Latex:
\mforall{}es:EO
\mforall{}[P1,P2,Q1,Q2:E {}\mrightarrow{} \mBbbP{}].
\mforall{}f:\{e:E| P1 e\} {}\mrightarrow{} \{e:E| Q1 e\}
((\mforall{}e:E. (P1 e \mLeftarrow{}{}\mRightarrow{} P2 e)) {}\mRightarrow{} (\mforall{}e:E. (Q1 e \mLeftarrow{}{}\mRightarrow{} Q2 e)) {}\mRightarrow{} (Q1 \mleftarrow{}\mleftarrow{}{} f{}{} P1 \mLeftarrow{}{}\mRightarrow{} Q2 \mleftarrow{}\mleftarrow{}{} f{}{} P2))
Date html generated:
2016_05_16-AM-10_28_11
Last ObjectModification:
2015_12_28-PM-09_19_45
Theory : new!event-ordering
Home
Index