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Theorem isphtpc 21367
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 4438 . . 3  |-  ( F (  ~=ph  `  J ) G  <->  <. F ,  G >.  e.  (  ~=ph  `  J
) )
2 df-phtpc 21365 . . . . 5  |-  ~=ph  =  ( j  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  j
)  /\  ( f
( PHtpy `  j )
g )  =/=  (/) ) } )
32dmmptss 5493 . . . 4  |-  dom  ~=ph  C_  Top
4 elfvdm 5882 . . . 4  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  dom  ~=ph 
)
53, 4sseldi 3487 . . 3  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  Top )
61, 5sylbi 195 . 2  |-  ( F (  ~=ph  `  J ) G  ->  J  e.  Top )
7 cntop2 19615 . . 3  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
873ad2ant1 1018 . 2  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) )  ->  J  e. 
Top )
9 oveq2 6289 . . . . . . . . 9  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
109sseq2d 3517 . . . . . . . 8  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  { f ,  g }  C_  ( II  Cn  J
) ) )
11 vex 3098 . . . . . . . . 9  |-  f  e. 
_V
12 vex 3098 . . . . . . . . 9  |-  g  e. 
_V
1311, 12prss 4169 . . . . . . . 8  |-  ( ( f  e.  ( II 
Cn  J )  /\  g  e.  ( II  Cn  J ) )  <->  { f ,  g }  C_  ( II  Cn  J
) )
1410, 13syl6bbr 263 . . . . . . 7  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  ( f  e.  ( II  Cn  J
)  /\  g  e.  ( II  Cn  J
) ) ) )
15 fveq2 5856 . . . . . . . . 9  |-  ( j  =  J  ->  ( PHtpy `  j )  =  ( PHtpy `  J )
)
1615oveqd 6298 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( PHtpy `  j
) g )  =  ( f ( PHtpy `  J ) g ) )
1716neeq1d 2720 . . . . . . 7  |-  ( j  =  J  ->  (
( f ( PHtpy `  j ) g )  =/=  (/)  <->  ( f (
PHtpy `  J ) g )  =/=  (/) ) )
1814, 17anbi12d 710 . . . . . 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 4500 . . . . 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 6309 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
2120, 20xpex 6589 . . . . . 6  |-  ( ( II  Cn  J )  X.  ( II  Cn  J ) )  e. 
_V
22 opabssxp 5064 . . . . . 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 4582 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }  e.  _V
2419, 2, 23fvmpt 5941 . . . 4  |-  ( J  e.  Top  ->  (  ~=ph  `  J )  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } )
2524breqd 4448 . . 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 6290 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( PHtpy `  J ) g )  =  ( F (
PHtpy `  J ) G ) )
2726neeq1d 2720 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f (
PHtpy `  J ) g )  =/=  (/)  <->  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
28 eqid 2443 . . . . 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 5065 . . . 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 976 . . . 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 252 . . 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 261 . 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 353 1  |-  ( F (  ~=ph  `  J ) G  <->  ( F  e.  ( II  Cn  J
)  /\  G  e.  ( II  Cn  J
)  /\  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638    C_ wss 3461   (/)c0 3770   {cpr 4016   <.cop 4020   class class class wbr 4437   {copab 4494    X. cxp 4987   dom cdm 4989   ` cfv 5578  (class class class)co 6281   Topctop 19267    Cn ccn 19598   IIcii 21252   PHtpycphtpy 21341    ~=ph cphtpc 21342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-map 7424  df-top 19272  df-topon 19275  df-cn 19601  df-phtpc 21365
This theorem is referenced by:  phtpcer  21368  phtpc01  21369  reparpht  21371  phtpcco2  21372  pcohtpylem  21392  pcohtpy  21393  pcorevlem  21399  pi1blem  21412  txsconlem  28558  txscon  28559  cvxscon  28561  cvmliftpht  28636
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