Theorem reldvdsr 20232

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 20229	. . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2	1	relmptopab 7590	. 2 ⊢ Rel (∥r‘𝑅)
3		reldvdsr.1	. . 3 ⊢ ∥ = (∥r‘𝑅)
4	3	releqi 5715	. 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅))
5	2, 4	mpbir 231	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3433  Rel wrel 5618  ‘cfv 6476  (class class class)co 7340  Basecbs 17107  ._rcmulr 17149  ∥_rcdsr 20226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fv 6484  df-dvdsr 20229

This theorem is referenced by:  dvdsr  20234  isunit  20245  subrgdvds  20455  ellpi  33306