Theorem reldvdsr 20488

Description: The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypothesis

Ref	Expression
reldvdsr.1	⊢ ∥ = (∥r‘𝑅)

Assertion

Ref	Expression
reldvdsr	⊢ Rel ∥

Proof of Theorem reldvdsr

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvdsr 20485	. . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2	1	relmptopab 7670	. 2 ⊢ Rel (∥r‘𝑅)
3		reldvdsr.1	. . 3 ⊢ ∥ = (∥r‘𝑅)
4	3	releqi 5766	. 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅))
5	2, 4	mpbir 234	1 ⊢ Rel ∥

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  Vcvv 3457  Rel wrel 5668  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  ._rcmulr 17333  ∥_rcdsr 20482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-dvdsr 20485

This theorem is used by:  dvdsr  20490  isunit  20501  subrgdvds  20735  ellpi  33751