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Theorem isphtpc 18976
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 4177 . . 3  |-  ( F (  ~=ph  `  J ) G  <->  <. F ,  G >.  e.  (  ~=ph  `  J
) )
2 df-phtpc 18974 . . . . 5  |-  ~=ph  =  ( j  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  j
)  /\  ( f
( PHtpy `  j )
g )  =/=  (/) ) } )
32dmmptss 5329 . . . 4  |-  dom  ~=ph  C_  Top
4 elfvdm 5720 . . . 4  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  dom  ~=ph 
)
53, 4sseldi 3310 . . 3  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  Top )
61, 5sylbi 188 . 2  |-  ( F (  ~=ph  `  J ) G  ->  J  e.  Top )
7 cntop2 17263 . . 3  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
873ad2ant1 978 . 2  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) )  ->  J  e. 
Top )
9 oveq2 6052 . . . . . . . . 9  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
109sseq2d 3340 . . . . . . . 8  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  { f ,  g }  C_  ( II  Cn  J
) ) )
11 vex 2923 . . . . . . . . 9  |-  f  e. 
_V
12 vex 2923 . . . . . . . . 9  |-  g  e. 
_V
1311, 12prss 3916 . . . . . . . 8  |-  ( ( f  e.  ( II 
Cn  J )  /\  g  e.  ( II  Cn  J ) )  <->  { f ,  g }  C_  ( II  Cn  J
) )
1410, 13syl6bbr 255 . . . . . . 7  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  ( f  e.  ( II  Cn  J
)  /\  g  e.  ( II  Cn  J
) ) ) )
15 fveq2 5691 . . . . . . . . 9  |-  ( j  =  J  ->  ( PHtpy `  j )  =  ( PHtpy `  J )
)
1615oveqd 6061 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( PHtpy `  j
) g )  =  ( f ( PHtpy `  J ) g ) )
1716neeq1d 2584 . . . . . . 7  |-  ( j  =  J  ->  (
( f ( PHtpy `  j ) g )  =/=  (/)  <->  ( f (
PHtpy `  J ) g )  =/=  (/) ) )
1814, 17anbi12d 692 . . . . . 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 4235 . . . . 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 6069 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
2120, 20xpex 4953 . . . . . 6  |-  ( ( II  Cn  J )  X.  ( II  Cn  J ) )  e. 
_V
22 opabssxp 4913 . . . . . 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 4312 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }  e.  _V
2419, 2, 23fvmpt 5769 . . . 4  |-  ( J  e.  Top  ->  (  ~=ph  `  J )  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } )
2524breqd 4187 . . 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 6053 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( PHtpy `  J ) g )  =  ( F (
PHtpy `  J ) G ) )
2726neeq1d 2584 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f (
PHtpy `  J ) g )  =/=  (/)  <->  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
28 eqid 2408 . . . . 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 4914 . . . 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 938 . . . 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 244 . . 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 253 . 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 343 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571    C_ wss 3284   (/)c0 3592   {cpr 3779   <.cop 3781   class class class wbr 4176   {copab 4229    X. cxp 4839   dom cdm 4841   ` cfv 5417  (class class class)co 6044   Topctop 16917    Cn ccn 17246   IIcii 18862   PHtpycphtpy 18950    ~=ph cphtpc 18951
This theorem is referenced by:  phtpcer  18977  phtpc01  18978  reparpht  18980  phtpcco2  18981  pcohtpylem  19001  pcohtpy  19002  pcorevlem  19008  pi1blem  19021  txsconlem  24884  txscon  24885  cvxscon  24887  cvmliftpht  24962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-top 16922  df-topon 16925  df-cn 17249  df-phtpc 18974
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