Theorem reldvdsr 20386

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 20383	. . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2	1	relmptopab 7700	. 2 ⊢ Rel (∥r‘𝑅)
3		reldvdsr.1	. . 3 ⊢ ∥ = (∥r‘𝑅)
4	3	releqi 5801	. 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅))
5	2, 4	mpbir 231	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  ._rcmulr 17312  ∥_rcdsr 20380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-dvdsr 20383

This theorem is referenced by:  dvdsr  20388  isunit  20399  subrgdvds  20614  ellpi  33366