Nuprl Lemma : weak-antecedent-surjection_functionality_wrt_pred_equiv
∀es:EO
∀[P1,P2,Q1,Q2:E ─→ ℙ]. ∀f:{e:E| P1 e} ─→ {e:E| Q1 e} . (P1 ⇐⇒ P2 ⇒ Q1 ⇐⇒ Q2 ⇒ (Q1 ←←= f== P1 ⇐⇒ Q2 ←←= f== P2))
Proof
Definitions occuring in Statement :
weak-antecedent-surjection: Q ←←= f== P,
es-E: E,
event_ordering: EO,
predicate_equivalent: P1 ⇐⇒ P2,
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]
Lemmas :
subtype_rel_sets,
equal_wf,
set_wf,
weak-antecedent-surjection_wf,
subtype_rel_dep_function,
es-E_wf,
all_wf,
iff_wf,
event_ordering_wf
\mforall{}es:EO
\mforall{}[P1,P2,Q1,Q2:E {}\mrightarrow{} \mBbbP{}].
\mforall{}f:\{e:E| P1 e\} {}\mrightarrow{} \{e:E| Q1 e\} . (P1 \mLeftarrow{}{}\mRightarrow{} P2 {}\mRightarrow{} Q1 \mLeftarrow{}{}\mRightarrow{} Q2 {}\mRightarrow{} (Q1 \mleftarrow{}\mleftarrow{}= f== P1 \mLeftarrow{}{}\mRightarrow{} Q2 \mleftarrow{}\mleftarrow{}= f== P2))
Date html generated:
2015_07_17-AM-09_02_28
Last ObjectModification:
2015_01_27-PM-00_55_44
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