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Definition df-phtpc 18452
Description: Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
Assertion
Ref Expression
df-phtpc  |-  ~=ph  =  ( x  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x
)  /\  ( f
( PHtpy `  x )
g )  =/=  (/) ) } )
Distinct variable group:    x, f, g

Detailed syntax breakdown of Definition df-phtpc
StepHypRef Expression
1 cphtpc 18429 . 2  class  ~=ph
2 vx . . 3  set  x
3 ctop 16593 . . 3  class  Top
4 vf . . . . . . . 8  set  f
54cv 1618 . . . . . . 7  class  f
6 vg . . . . . . . 8  set  g
76cv 1618 . . . . . . 7  class  g
85, 7cpr 3615 . . . . . 6  class  { f ,  g }
9 cii 18341 . . . . . . 7  class  II
102cv 1618 . . . . . . 7  class  x
11 ccn 16916 . . . . . . 7  class  Cn
129, 10, 11co 5792 . . . . . 6  class  ( II 
Cn  x )
138, 12wss 3127 . . . . 5  wff  { f ,  g }  C_  ( II  Cn  x
)
14 cphtpy 18428 . . . . . . . 8  class  PHtpy
1510, 14cfv 4673 . . . . . . 7  class  ( PHtpy `  x )
165, 7, 15co 5792 . . . . . 6  class  ( f ( PHtpy `  x )
g )
17 c0 3430 . . . . . 6  class  (/)
1816, 17wne 2421 . . . . 5  wff  ( f ( PHtpy `  x )
g )  =/=  (/)
1913, 18wa 360 . . . 4  wff  ( { f ,  g } 
C_  ( II  Cn  x )  /\  (
f ( PHtpy `  x
) g )  =/=  (/) )
2019, 4, 6copab 4050 . . 3  class  { <. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x )  /\  (
f ( PHtpy `  x
) g )  =/=  (/) ) }
212, 3, 20cmpt 4051 . 2  class  ( x  e.  Top  |->  { <. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x )  /\  (
f ( PHtpy `  x
) g )  =/=  (/) ) } )
221, 21wceq 1619 1  wff  ~=ph  =  ( x  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x
)  /\  ( f
( PHtpy `  x )
g )  =/=  (/) ) } )
Colors of variables: wff set class
This definition is referenced by:  phtpcrel  18453  isphtpc  18454
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