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Theorem isphtpc 20708
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 4404 . . 3  |-  ( F (  ~=ph  `  J ) G  <->  <. F ,  G >.  e.  (  ~=ph  `  J
) )
2 df-phtpc 20706 . . . . 5  |-  ~=ph  =  ( j  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  j
)  /\  ( f
( PHtpy `  j )
g )  =/=  (/) ) } )
32dmmptss 5445 . . . 4  |-  dom  ~=ph  C_  Top
4 elfvdm 5828 . . . 4  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  dom  ~=ph 
)
53, 4sseldi 3465 . . 3  |-  ( <. F ,  G >.  e.  (  ~=ph  `  J )  ->  J  e.  Top )
61, 5sylbi 195 . 2  |-  ( F (  ~=ph  `  J ) G  ->  J  e.  Top )
7 cntop2 18987 . . 3  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
873ad2ant1 1009 . 2  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J
) G )  =/=  (/) )  ->  J  e. 
Top )
9 oveq2 6211 . . . . . . . . 9  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
109sseq2d 3495 . . . . . . . 8  |-  ( j  =  J  ->  ( { f ,  g }  C_  ( II  Cn  j )  <->  { f ,  g }  C_  ( II  Cn  J
) ) )
11 vex 3081 . . . . . . . . 9  |-  f  e. 
_V
12 vex 3081 . . . . . . . . 9  |-  g  e. 
_V
1311, 12prss 4138 . . . . . . . 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 5802 . . . . . . . . 9  |-  ( j  =  J  ->  ( PHtpy `  j )  =  ( PHtpy `  J )
)
1615oveqd 6220 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( PHtpy `  j
) g )  =  ( f ( PHtpy `  J ) g ) )
1716neeq1d 2729 . . . . . . 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 4466 . . . . 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 6228 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
2120, 20xpex 6621 . . . . . 6  |-  ( ( II  Cn  J )  X.  ( II  Cn  J ) )  e. 
_V
22 opabssxp 5022 . . . . . 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 4548 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J ) )  /\  ( f ( PHtpy `  J )
g )  =/=  (/) ) }  e.  _V
2419, 2, 23fvmpt 5886 . . . 4  |-  ( J  e.  Top  ->  (  ~=ph  `  J )  =  { <. f ,  g >.  |  ( ( f  e.  ( II  Cn  J )  /\  g  e.  ( II  Cn  J
) )  /\  (
f ( PHtpy `  J
) g )  =/=  (/) ) } )
2524breqd 4414 . . 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 6212 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( PHtpy `  J ) g )  =  ( F (
PHtpy `  J ) G ) )
2726neeq1d 2729 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f (
PHtpy `  J ) g )  =/=  (/)  <->  ( F
( PHtpy `  J ) G )  =/=  (/) ) )
28 eqid 2454 . . . . 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 5023 . . . 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 967 . . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648    C_ wss 3439   (/)c0 3748   {cpr 3990   <.cop 3994   class class class wbr 4403   {copab 4460    X. cxp 4949   dom cdm 4951   ` cfv 5529  (class class class)co 6203   Topctop 18640    Cn ccn 18970   IIcii 20593   PHtpycphtpy 20682    ~=ph cphtpc 20683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-top 18645  df-topon 18648  df-cn 18973  df-phtpc 20706
This theorem is referenced by:  phtpcer  20709  phtpc01  20710  reparpht  20712  phtpcco2  20713  pcohtpylem  20733  pcohtpy  20734  pcorevlem  20740  pi1blem  20753  txsconlem  27296  txscon  27297  cvxscon  27299  cvmliftpht  27374
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