Nuprl Lemma : biject_functionality
∀[A1,B1,A2,B2:Type]. ∀f:A1 ⟶ B1. (Bij(A1;B1;f) ⇐⇒ Bij(A2;B2;f)) supposing (B1 ≡ B2 and A1 ≡ A2)
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
guard: {T},
prop: ℙ,
surject: Surj(A;B;f),
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
biject_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
rename,
independent_pairFormation,
promote_hyp,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
extract_by_obid,
isectElimination,
cumulativity,
functionExtensionality,
because_Cache,
dependent_pairFormation,
productEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A1,B1,A2,B2:Type]. \mforall{}f:A1 {}\mrightarrow{} B1. (Bij(A1;B1;f) \mLeftarrow{}{}\mRightarrow{} Bij(A2;B2;f)) supposing (B1 \mequiv{} B2 and A1 \mequiv{} A2)
Date html generated:
2018_05_21-PM-06_33_11
Last ObjectModification:
2017_07_26-PM-04_52_04
Theory : general
Home
Index