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Theorem phtpcrel 22102
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 22101 . 2  |-  ~=ph  =  ( x  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x
)  /\  ( f
( PHtpy `  x )
g )  =/=  (/) ) } )
21relmptopab 6536 1  |-  Rel  (  ~=ph  `  J )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    =/= wne 2641    C_ wss 3390   (/)c0 3722   {cpr 3961   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   Topctop 19994    Cn ccn 20317   IIcii 21985   PHtpycphtpy 22077    ~=ph cphtpc 22078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-phtpc 22101
This theorem is referenced by:  phtpcer  22104
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