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
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