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Theorem phtpcrel 24939
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 ( ≃ph𝐽)

Proof of Theorem phtpcrel
Dummy variables 𝑥 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-phtpc 24938 . 2 ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
21relmptopab 7677 1 Rel ( ≃ph𝐽)
Colors of variables: wff setvar class
Syntax hints:  wa 394  wne 2937  wss 3949  c0 4326  {cpr 4634  Rel wrel 5687  cfv 6553  (class class class)co 7426  Topctop 22815   Cn ccn 23148  IIcii 24815  PHtpycphtpy 24914  phcphtpc 24915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-phtpc 24938
This theorem is referenced by:  phtpcer  24941
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