Theorem reldvdsr 20396

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 20393	. . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
2	1	relmptopab 7641	. 2 ⊢ Rel (∥r‘𝑅)
3		reldvdsr.1	. . 3 ⊢ ∥ = (∥r‘𝑅)
4	3	releqi 5746	. 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅))
5	2, 4	mpbir 233	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  Vcvv 3453  Rel wrel 5648  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  ._rcmulr 17278  ∥_rcdsr 20390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fv 6524  df-dvdsr 20393

This theorem is referenced by:  dvdsr  20398  isunit  20409  subrgdvds  20623  ellpi  33520