MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isphtpc Unicode version

Theorem isphtpc 18596
Description: The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)
Assertion
Ref Expression
isphtpc  |-  ( F (  ~=ph  `  J ) G  <->  ( F  e.  ( II  Cn  J
)  /\  G  e.  ( II  Cn  J
)  /\  ( F
( PHtpy `  J ) G )  =/=  (/) ) )

Proof of Theorem isphtpc
Dummy variables  f 
g  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4105 . . 3  |-  ( F (  ~=ph  `  J ) G  <->  <. F ,  G >.  e.  (  ~=ph  `  J
) )
2 df-phtpc 18594 . . . . 5  |-  ~=ph  =  ( j  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  j
)  /\  ( f
( PHtpy `  j )
g )  =/=  (/) ) } )
32dmmptss 5251 . . . 4  |-  dom  ~=ph  C_  Top
4 elfvdm 5637 . . . 4  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  dom  ~=ph 
)
53, 4sseldi 3254 . . 3  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  Top )
61, 5sylbi 187 . 2  |-  ( F (  ~=ph  `  J ) G  ->  J  e.  Top )
7 cntop2 17077 . . 3  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
873ad2ant1 976 . 2  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) )  ->  J  e. 
Top )
9 oveq2 5953 . . . . . . . . 9  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
109sseq2d 3282 . . . . . . . 8  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  { f ,  g }  C_  ( II  Cn  J
) ) )
11 vex 2867 . . . . . . . . 9  |-  f  e. 
_V
12 vex 2867 . . . . . . . . 9  |-  g  e. 
_V
1311, 12prss 3848 . . . . . . . 8  |-  ( ( f  e.  ( II 
Cn  J )  /\  g  e.  ( II  Cn  J ) )  <->  { f ,  g }  C_  ( II  Cn  J
) )
1410, 13syl6bbr 254 . . . . . . 7  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  ( f  e.  ( II  Cn  J
)  /\  g  e.  ( II  Cn  J
) ) ) )
15 fveq2 5608 . . . . . . . . 9  |-  ( j  =  J  ->  ( PHtpy `  j )  =  ( PHtpy `  J )
)
1615oveqd 5962 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( PHtpy `  j
) g )  =  ( f ( PHtpy `  J ) g ) )
1716neeq1d 2534 . . . . . . 7  |-  ( j  =  J  ->  (
( f ( PHtpy `  j ) g )  =/=  (/)  <->  ( f (
PHtpy `  J ) g )  =/=  (/) ) )
1814, 17anbi12d 691 . . . . . 6  |-  ( j  =  J  ->  (
( { f ,  g }  C_  (
II  Cn  j )  /\  ( f ( PHtpy `  j ) g )  =/=  (/) )  <->  ( (
f  e.  ( II 
Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J ) g )  =/=  (/) ) ) )
1918opabbidv 4163 . . . . 5  |-  ( j  =  J  ->  { <. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  j )  /\  (
f ( PHtpy `  j
) g )  =/=  (/) ) }  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } )
20 ovex 5970 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
2120, 20xpex 4883 . . . . . 6  |-  ( ( II  Cn  J )  X.  ( II  Cn  J ) )  e. 
_V
22 opabssxp 4844 . . . . . 6  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) } 
C_  ( ( II 
Cn  J )  X.  ( II  Cn  J
) )
2321, 22ssexi 4240 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }  e.  _V
2419, 2, 23fvmpt 5685 . . . 4  |-  ( J  e.  Top  ->  (  ~=ph  `  J )  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } )
2524breqd 4115 . . 3  |-  ( J  e.  Top  ->  ( F (  ~=ph  `  J
) G  <->  F { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } G ) )
26 oveq12 5954 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( PHtpy `  J ) g )  =  ( F (
PHtpy `  J ) G ) )
2726neeq1d 2534 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f (
PHtpy `  J ) g )  =/=  (/)  <->  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
28 eqid 2358 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }
2927, 28brab2ga 4845 . . . 4  |-  ( F { <. f ,  g
>.  |  ( (
f  e.  ( II 
Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J ) g )  =/=  (/) ) } G  <->  ( ( F  e.  ( II  Cn  J )  /\  G  e.  ( II  Cn  J ) )  /\  ( F ( PHtpy `  J ) G )  =/=  (/) ) )
30 df-3an 936 . . . 4  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) )  <->  ( ( F  e.  ( II  Cn  J )  /\  G  e.  ( II  Cn  J
) )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) ) )
3129, 30bitr4i 243 . . 3  |-  ( F { <. f ,  g
>.  |  ( (
f  e.  ( II 
Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J ) g )  =/=  (/) ) } G  <->  ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) ) )
3225, 31syl6bb 252 . 2  |-  ( J  e.  Top  ->  ( F (  ~=ph  `  J
) G  <->  ( F  e.  ( II  Cn  J
)  /\  G  e.  ( II  Cn  J
)  /\  ( F
( PHtpy `  J ) G )  =/=  (/) ) ) )
336, 8, 32pm5.21nii 342 1  |-  ( F (  ~=ph  `  J ) G  <->  ( F  e.  ( II  Cn  J
)  /\  G  e.  ( II  Cn  J
)  /\  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521    C_ wss 3228   (/)c0 3531   {cpr 3717   <.cop 3719   class class class wbr 4104   {copab 4157    X. cxp 4769   dom cdm 4771   ` cfv 5337  (class class class)co 5945   Topctop 16737    Cn ccn 17060   IIcii 18482   PHtpycphtpy 18570    ~=ph cphtpc 18571
This theorem is referenced by:  phtpcer  18597  phtpc01  18598  reparpht  18600  phtpcco2  18601  pcohtpylem  18621  pcohtpy  18622  pcorevlem  18628  pi1blem  18641  txsconlem  24175  txscon  24176  cvxscon  24178  cvmliftpht  24253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-map 6862  df-top 16742  df-topon 16745  df-cn 17063  df-phtpc 18594
  Copyright terms: Public domain W3C validator