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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxkococnlem 17301* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  F  =  ( f  e.  ( S  Cn  T ) ,  g  e.  ( R  Cn  S )  |->  ( f  o.  g ) )   &    |-  ( ph  ->  S  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  K  C_  U. R )   &    |-  ( ph  ->  ( Rt  K )  e.  Comp )   &    |-  ( ph  ->  V  e.  T )   &    |-  ( ph  ->  A  e.  ( S  Cn  T ) )   &    |-  ( ph  ->  B  e.  ( R  Cn  S ) )   &    |-  ( ph  ->  ( ( A  o.  B ) " K )  C_  V )   =>    |-  ( ph  ->  E. z  e.  ( ( T  ^ k o  S )  tX  ( S  ^ k o  R ) ) (
 <. A ,  B >.  e.  z  /\  z  C_  ( `' F " { h  e.  ( R  Cn  T )  |  ( h " K )  C_  V } ) ) )
 
Theoremxkococn 17302* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  F  =  ( f  e.  ( S  Cn  T ) ,  g  e.  ( R  Cn  S )  |->  ( f  o.  g ) )   =>    |-  ( ( R  e.  Top  /\  S  e. 𝑛Locally  Comp  /\  T  e.  Top )  ->  F  e.  ( ( ( T  ^ k o  S )  tX  ( S  ^ k o  R ) )  Cn  ( T  ^ k o  R ) ) )
 
11.1.17  Continuous function-builders
 
Theoremcnmptid 17303* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  x )  e.  ( J  Cn  J ) )
 
Theoremcnmptc 17304* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  P  e.  Y )   =>    |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
 
Theoremcnmpt11 17305* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  B )  e.  ( K  Cn  L ) )   &    |-  ( y  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmpt11f 17306* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  F  e.  ( K  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `  A ) )  e.  ( J  Cn  L ) )
 
Theoremcnmpt1t 17307* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  <. A ,  B >. )  e.  ( J  Cn  ( K  tX  L ) ) )
 
Theoremcnmpt12f 17308* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  F  e.  ( ( K 
 tX  L )  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
 
Theoremcnmpt12 17309* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  ( y  e.  Y ,  z  e.  Z  |->  C )  e.  ( ( K 
 tX  L )  Cn  M ) )   &    |-  (
 ( y  =  A  /\  z  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  X  |->  D )  e.  ( J  Cn  M ) )
 
Theoremcnmpt1st 17310* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  x )  e.  ( ( J 
 tX  K )  Cn  J ) )
 
Theoremcnmpt2nd 17311* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  y )  e.  ( ( J 
 tX  K )  Cn  K ) )
 
Theoremcnmpt2c 17312* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  P  e.  Z )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  P )  e.  ( ( J 
 tX  K )  Cn  L ) )
 
Theoremcnmpt21 17313* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  ( z  e.  Z  |->  B )  e.  ( L  Cn  M ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  C )  e.  ( ( J 
 tX  K )  Cn  M ) )
 
Theoremcnmpt21f 17314* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  F  e.  ( L  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( F `  A ) )  e.  ( ( J  tX  K )  Cn  M ) )
 
Theoremcnmpt2t 17315* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  <. A ,  B >. )  e.  (
 ( J  tX  K )  Cn  ( L  tX  M ) ) )
 
Theoremcnmpt22 17316* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  M  e.  (TopOn `  W ) )   &    |-  ( ph  ->  ( z  e.  Z ,  w  e.  W  |->  C )  e.  ( ( L 
 tX  M )  Cn  N ) )   &    |-  (
 ( z  =  A  /\  w  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  D )  e.  ( ( J 
 tX  K )  Cn  N ) )
 
Theoremcnmpt22f 17317* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   &    |-  ( ph  ->  F  e.  ( ( L 
 tX  M )  Cn  N ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J 
 tX  K )  Cn  N ) )
 
Theoremcnmpt1res 17318* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y  |->  A )  e.  ( K  Cn  L ) )
 
Theoremcnmpt2res 17319* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  N  =  ( Mt  W )   &    |-  ( ph  ->  M  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  W 
 C_  Z )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Z  |->  A )  e.  ( ( J  tX  M )  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y ,  y  e.  W  |->  A )  e.  ( ( K 
 tX  N )  Cn  L ) )
 
Theoremcnmptcom 17320* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   =>    |-  ( ph  ->  ( y  e.  Y ,  x  e.  X  |->  A )  e.  ( ( K 
 tX  J )  Cn  L ) )
 
Theoremcnmptkc 17321* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  x ) )  e.  ( J  Cn  ( J  ^ k o  K )
 ) )
 
Theoremcnmptkp 17322* The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  B  e.  Y )   &    |-  (
 y  =  B  ->  A  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmptk1 17323* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  ( z  e.  Z  |->  B )  e.  ( L  Cn  M ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  e.  ( J  Cn  ( M  ^ k o  K ) ) )
 
Theoremcnmpt1k 17324* The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  M  e.  (TopOn `  W )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  ( z  e.  Z  |->  B ) )  e.  ( K  Cn  ( M  ^ k o  L ) ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( y  e.  Y  |->  ( x  e.  X  |->  C ) )  e.  ( K  Cn  ( M  ^ k o  J ) ) )
 
Theoremcnmptkk 17325* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  M  e.  (TopOn `  W )
 )   &    |-  ( ph  ->  L  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  ( x  e.  X  |->  ( z  e.  Z  |->  B ) )  e.  ( J  Cn  ( M  ^ k o  L )
 ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  e.  ( J  Cn  ( M  ^ k o  K )
 ) )
 
Theoremxkofvcn 17326* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17298.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   &    |-  F  =  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `  x ) )   =>    |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  F  e.  ( ( ( S  ^ k o  R )  tX  R )  Cn  S ) )
 
Theoremcnmptk1p 17327* The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  K  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   &    |-  ( y  =  B  ->  A  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmptk2 17328* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  K  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )
 
Theoremxkoinjcn 17329* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  F  =  ( x  e.  X  |->  ( y  e.  Y  |->  <. y ,  x >. ) )   =>    |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  F  e.  ( R  Cn  ( ( S  tX  R )  ^ k o  S ) ) )
 
Theoremcnmpt2k 17330* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ k o  K )
 ) )
 
Theoremtxcon 17331 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( ( R  e.  Con  /\  S  e.  Con )  ->  ( R  tX  S )  e.  Con )
 
11.1.18  Quotient maps and quotient topology
 
Syntaxckq 17332 Extend class notation with the Kolmogorov quotient function.
 class KQ
 
Definitiondf-kq 17333* Define the Kolmogorov quotient. This is a function on topologies which maps a topology to its quotient under the topological distinguishability map, which takes a point to the set of open sets that contain it. Two points are mapped to the same image under this function iff they are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |- KQ 
 =  ( j  e. 
 Top  |->  ( j qTop  ( x  e.  U. j  |->  { y  e.  j  |  x  e.  y }
 ) ) )
 
Theoremqtopval 17334* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F  e.  W ) 
 ->  ( J qTop  F )  =  { s  e. 
 ~P ( F " X )  |  (
 ( `' F "
 s )  i^i  X )  e.  J }
 )
 
Theoremqtopval2 17335* Value of the quotient topology function when  F is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ( J qTop  F )  =  { s  e. 
 ~P Y  |  ( `' F " s )  e.  J } )
 
Theoremelqtop 17336 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
 
Theoremqtopres 17337 The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that  F be a function with domain  X. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
 
Theoremqtoptop2 17338 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( J  e.  Top  /\  F  e.  V  /\  Fun 
 F )  ->  ( J qTop  F )  e.  Top )
 
Theoremqtoptop 17339 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e.  Top )
 
Theoremelqtop2 17340 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F : X -onto-> Y )  ->  ( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
 
Theoremqtopuni 17341 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F : X -onto-> Y )  ->  Y  =  U. ( J qTop  F ) )
 
Theoremelqtop3 17342 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y ) 
 ->  ( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
 
Theoremqtoptopon 17343 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y ) 
 ->  ( J qTop  F )  e.  (TopOn `  Y ) )
 
Theoremqtopid 17344 A quotient map a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
 
Theoremidqtop 17345 The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J qTop  (  _I  |`  X ) )  =  J )
 
Theoremqtopcmplem 17346 Lemma for qtopcmp 17347 and qtopcon 17348. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   &    |-  ( J  e.  A  ->  J  e.  Top )   &    |-  (
 ( J  e.  A  /\  F : X -onto-> U. ( J qTop  F )  /\  F  e.  ( J  Cn  ( J qTop  F ) ) )  ->  ( J qTop  F )  e.  A )   =>    |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )
 
Theoremqtopcmp 17347 A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Comp  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e.  Comp )
 
Theoremqtopcon 17348 A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Con  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e.  Con )
 
Theoremqtopkgen 17349 A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e. 
 ran 𝑘Gen )
 
Theorembasqtop 17350 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  TopBases  /\  F : X -1-1-onto-> Y )  ->  ( J qTop  F )  e.  TopBases )
 
Theoremtgqtop 17351 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  TopBases  /\  F : X -1-1-onto-> Y )  ->  (
 ( topGen `  J ) qTop  F )  =  ( topGen `  ( J qTop  F ) ) )
 
Theoremqtopcld 17352 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y ) 
 ->  ( A  e.  ( Clsd `  ( J qTop  F ) )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  ( Clsd `  J ) ) ) )
 
Theoremqtopcn 17353 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  ->  ( G  e.  ( ( J qTop  F )  Cn  K )  <->  ( G  o.  F )  e.  ( J  Cn  K ) ) )
 
Theoremqtopss 17354 A surjective continuous function from  J to  K induces a topology  J qTop  F on the base set of  K. This topology is in general finer than  K. Together with qtopid 17344, this implies that  J qTop  F is the finest topology making  F continuous, i.e. the final topology with respect to the family  { F }. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ( F  e.  ( J  Cn  K ) 
 /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
 
Theoremqtopeu 17355* Universal property of the quotient topology. If  G is a function from  J to  K which is equal on all equivalent elements under  F, then there is a unique continuous map  f : ( J  /  F ) --> K such that  G  =  f  o.  F, and we say that  G "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  ( F `
  x )  =  ( F `  y
 ) ) )  ->  ( G `  x )  =  ( G `  y ) )   =>    |-  ( ph  ->  E! f  e.  ( ( J qTop  F )  Cn  K ) G  =  ( f  o.  F ) )
 
Theoremqtoprest 17356 If  A is a saturated open or closed set (where saturated means that  A  =  ( `' F " U ) for some  U), then the restriction of the quotient map  F to  A is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  U 
 C_  Y )   &    |-  ( ph  ->  A  =  ( `' F " U ) )   &    |-  ( ph  ->  ( A  e.  J  \/  A  e.  ( Clsd `  J ) ) )   =>    |-  ( ph  ->  ( ( J qTop  F )t  U )  =  ( ( Jt  A ) qTop  ( F  |`  A ) ) )
 
Theoremqtopomap 17357* If  F is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ran 
 F  =  Y )   &    |-  ( ( ph  /\  x  e.  J )  ->  ( F " x )  e.  K )   =>    |-  ( ph  ->  K  =  ( J qTop  F ) )
 
Theoremqtopcmap 17358* If  F is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ran 
 F  =  Y )   &    |-  ( ( ph  /\  x  e.  ( Clsd `  J )
 )  ->  ( F " x )  e.  ( Clsd `  K ) )   =>    |-  ( ph  ->  K  =  ( J qTop  F ) )
 
Theoremimastopn 17359 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  W )   &    |-  J  =  ( TopOpen `  R )   &    |-  O  =  ( TopOpen `  U )   =>    |-  ( ph  ->  O  =  ( J qTop  F ) )
 
Theoremimastps 17360 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  TopSp )   =>    |-  ( ph  ->  U  e.  TopSp )
 
Theoremdivstps 17361 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  E )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  E  e.  W )   &    |-  ( ph  ->  R  e.  TopSp )   =>    |-  ( ph  ->  U  e.  TopSp )
 
Theoremkqfval 17362* Value of the function appearing in df-kq 17333. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  V  /\  A  e.  X )  ->  ( F `  A )  =  { y  e.  J  |  A  e.  y } )
 
Theoremkqfeq 17363* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  A. y  e.  J  ( A  e.  y  <->  B  e.  y ) ) )
 
Theoremkqffn 17364* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  V  ->  F  Fn  X )
 
Theoremkqval 17365* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  (KQ `  J )  =  ( J qTop  F ) )
 
Theoremkqtopon 17366* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  (KQ `  J )  e.  (TopOn `  ran  F ) )
 
Theoremkqid 17367* The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  F  e.  ( J  Cn  (KQ `  J )
 ) )
 
Theoremist0-4 17368* The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Kol2  <->  F : X -1-1-> _V ) )
 
Theoremkqfvima 17369* When the image set is open, the quotient map satisfies a partial converse to fnfvima 5676, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  X ) 
 ->  ( A  e.  U  <->  ( F `  A )  e.  ( F " U ) ) )
 
Theoremkqsat 17370* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17356). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F " U ) )  =  U )
 
Theoremkqdisj 17371* A version of imain 5252 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( ( F
 " U )  i^i  ( F " ( A  \  U ) ) )  =  (/) )
 
Theoremkqcldsat 17372* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17356). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J ) ) 
 ->  ( `' F "
 ( F " U ) )  =  U )
 
Theoremkqopn 17373* The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  (KQ `  J ) )
 
Theoremkqcld 17374* The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J ) ) 
 ->  ( F " U )  e.  ( Clsd `  (KQ `  J )
 ) )
 
Theoremkqt0lem 17375* Lemma for kqt0 17385. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  (KQ `  J )  e. 
 Kol2 )
 
Theoremisr0 17376* The property " J is an R0 space". A space is R0 if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains  x also contains  y, so there is no separation, then  x and  y are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R0 if and only if its Kolmogorov quotient is T1, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  ( (KQ `  J )  e.  Fre  <->  A. z  e.  X  A. w  e.  X  (
 A. o  e.  J  ( z  e.  o  ->  w  e.  o ) 
 ->  A. o  e.  J  ( z  e.  o  <->  w  e.  o ) ) ) )
 
Theoremr0cld 17377* The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from  A is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { z  e.  X  |  A. o  e.  J  ( z  e.  o  <->  A  e.  o
 ) }  e.  ( Clsd `  J ) )
 
Theoremregr1lem 17378* Lemma for regr1 17389. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  J  e.  Reg )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  -.  E. m  e.  (KQ `  J ) E. n  e.  (KQ `  J ) ( ( F `  A )  e.  m  /\  ( F `  B )  e.  n  /\  ( m  i^i  n )  =  (/) ) )   =>    |-  ( ph  ->  ( A  e.  U  ->  B  e.  U ) )
 
Theoremregr1lem2 17379* A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Reg )  ->  (KQ `  J )  e.  Haus )
 
Theoremkqreglem1 17380* A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Reg )  ->  (KQ `  J )  e.  Reg )
 
Theoremkqreglem2 17381* If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Reg )  ->  J  e.  Reg )
 
Theoremkqnrmlem1 17382* A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Nrm )
 
Theoremkqnrmlem2 17383* If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Nrm )
 
Theoremkqtop 17384 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Top  <->  (KQ `  J )  e.  Top )
 
Theoremkqt0 17385 The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Top  <->  (KQ `  J )  e.  Kol2 )
 
Theoremkqf 17386 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |- KQ
 : Top --> Kol2
 
Theoremr0sep 17387* The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A. o  e.  J  ( A  e.  o  <->  B  e.  o ) ) )
 
Theoremnrmr0reg 17388 A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  (KQ `  J )  e.  Fre )  ->  J  e.  Reg )
 
Theoremregr1 17389 A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Reg  ->  (KQ `  J )  e. 
 Haus )
 
Theoremkqreg 17390 The Kolmogorov quotient of a regular space is regular. By regr1 17389 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Reg  <->  (KQ `  J )  e.  Reg )
 
Theoremkqnrm 17391 The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Nrm  <->  (KQ `  J )  e.  Nrm )
 
11.1.19  Homeomorphisms
 
Syntaxchmeo 17392 Extend class notation with the class of all homeomorphisms.
 class  Homeo
 
Syntaxchmph 17393 Extend class notation with the relation "is homeomorph to.".
 class  ~=
 
Definitiondf-hmeo 17394* Function returning all the homeomorphisms from topology  j to topology  k. (Contributed by FL, 14-Feb-2007.)
 |- 
 Homeo  =  ( j  e.  Top ,  k  e. 
 Top  |->  { f  e.  (
 j  Cn  k )  |  `' f  e.  (
 k  Cn  j ) } )
 
Definitiondf-hmph 17395 Definition of the relation  x is homeomorph to  y. (Contributed by FL, 14-Feb-2007.)
 |- 
 ~=  =  ( `' 
 Homeo  " ( _V  \  1o ) )
 
Theoremhmeofn 17396 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |- 
 Homeo  Fn  ( Top  X.  Top )
 
Theoremhmeofval 17397* The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  Homeo  K )  =  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
 
Theoremishmeo 17398 The predicate F is a homeomorphism between topology  J and topology  K. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( F  e.  ( J  Homeo  K )  <->  ( F  e.  ( J  Cn  K ) 
 /\  `' F  e.  ( K  Cn  J ) ) )
 
Theoremhmeocn 17399 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K ) )
 
Theoremhmeocnvcn 17400 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J ) )
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