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Theorem isphtpc 18492
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 4024 . . 3  |-  ( F (  ~=ph  `  J ) G  <->  <. F ,  G >.  e.  (  ~=ph  `  J
) )
2 df-phtpc 18490 . . . . 5  |-  ~=ph  =  ( j  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  j
)  /\  ( f
( PHtpy `  j )
g )  =/=  (/) ) } )
32dmmptss 5169 . . . 4  |-  dom  ~=ph  C_  Top
4 elfvdm 5554 . . . 4  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  dom  ~=ph 
)
53, 4sseldi 3178 . . 3  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  Top )
61, 5sylbi 187 . 2  |-  ( F (  ~=ph  `  J ) G  ->  J  e.  Top )
7 cntop2 16971 . . 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 5866 . . . . . . . . 9  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
109sseq2d 3206 . . . . . . . 8  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  { f ,  g }  C_  ( II  Cn  J
) ) )
11 vex 2791 . . . . . . . . 9  |-  f  e. 
_V
12 vex 2791 . . . . . . . . 9  |-  g  e. 
_V
1311, 12prss 3769 . . . . . . . 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 5525 . . . . . . . . 9  |-  ( j  =  J  ->  ( PHtpy `  j )  =  ( PHtpy `  J )
)
1615oveqd 5875 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( PHtpy `  j
) g )  =  ( f ( PHtpy `  J ) g ) )
1716neeq1d 2459 . . . . . . 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 4082 . . . . 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 5883 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
2120, 20xpex 4801 . . . . . 6  |-  ( ( II  Cn  J )  X.  ( II  Cn  J ) )  e. 
_V
22 opabssxp 4762 . . . . . 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 4159 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }  e.  _V
2419, 2, 23fvmpt 5602 . . . 4  |-  ( J  e.  Top  ->  (  ~=ph  `  J )  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } )
2524breqd 4034 . . 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 5867 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( PHtpy `  J ) g )  =  ( F (
PHtpy `  J ) G ) )
2726neeq1d 2459 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f (
PHtpy `  J ) g )  =/=  (/)  <->  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
28 eqid 2283 . . . . 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 4763 . . . 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 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   {cpr 3641   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Topctop 16631    Cn ccn 16954   IIcii 18379   PHtpycphtpy 18466    ~=ph cphtpc 18467
This theorem is referenced by:  phtpcer  18493  phtpc01  18494  reparpht  18496  phtpcco2  18497  pcohtpylem  18517  pcohtpy  18518  pcorevlem  18524  pi1blem  18537  txsconlem  23771  txscon  23772  cvxscon  23774  cvmliftpht  23849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-phtpc 18490
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