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Theorem phtpcrel 18891
Description: The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
Assertion
Ref Expression
phtpcrel  |-  Rel  (  ~=ph  `  J )

Proof of Theorem phtpcrel
Dummy variables  x  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-phtpc 18890 . 2  |-  ~=ph  =  ( x  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x
)  /\  ( f
( PHtpy `  x )
g )  =/=  (/) ) } )
21relmptopab 6233 1  |-  Rel  (  ~=ph  `  J )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    =/= wne 2552    C_ wss 3265   (/)c0 3573   {cpr 3760   Rel wrel 4825   ` cfv 5396  (class class class)co 6022   Topctop 16883    Cn ccn 17212   IIcii 18778   PHtpycphtpy 18866    ~=ph cphtpc 18867
This theorem is referenced by:  phtpcer  18893
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fv 5404  df-phtpc 18890
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