Theorem phtpcrel 24740

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.

Step	Hyp	Ref	Expression
1		df-phtpc 24739	. 2 ⊢ ≃ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
2	1	relmptopab 7660	1 ⊢ Rel ( ≃ph‘𝐽)

Syntax hints:   ∧ wa 395   ≠ wne 2939   ⊆ wss 3948  ∅c0 4322  {cpr 4630  Rel wrel 5681  ‘cfv 6543  (class class class)co 7412  Topctop 22616   Cn ccn 22949  IIcii 24616  PHtpycphtpy 24715   ≃_phcphtpc 24716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-phtpc 24739

This theorem is referenced by:  phtpcer  24742