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]
Lemmas :
subtype_rel_sets,
equal_wf,
set_wf,
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\}
((\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:
2015_07_17-AM-09_06_37
Last ObjectModification:
2015_01_27-PM-00_49_00
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