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Theorem phtpcrel 19010
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 19009 . 2  |-  ~=ph  =  ( x  e.  Top  |->  {
<. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x
)  /\  ( f
( PHtpy `  x )
g )  =/=  (/) ) } )
21relmptopab 6284 1  |-  Rel  (  ~=ph  `  J )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    =/= wne 2598    C_ wss 3312   (/)c0 3620   {cpr 3807   Rel wrel 4875   ` cfv 5446  (class class class)co 6073   Topctop 16950    Cn ccn 17280   IIcii 18897   PHtpycphtpy 18985    ~=ph cphtpc 18986
This theorem is referenced by:  phtpcer  19012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-phtpc 19009
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