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]
Lemmas :
subtype_rel_sets,
set_wf,
es-E_wf
\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:
2015_07_17-AM-09_06_23
Last ObjectModification:
2015_01_27-PM-00_48_54
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