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]
Definitions unfolded in proof :
predicate_equivalent: P1 ⇐⇒ P2,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
weak-antecedent-surjection: Q ←←= f== P,
weak-antecedent-function: Q ←==f== P,
member: t ∈ T,
cand: A c∧ B,
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],
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\} . (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:
2016_05_16-AM-10_17_22
Last ObjectModification:
2015_12_28-PM-09_25_02
Theory : new!event-ordering
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