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Theorem dvdsrval 19663
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
dvdsr.1 𝐵 = (Base‘𝑅)
dvdsr.2 = (∥r𝑅)
dvdsr.3 · = (.r𝑅)
Assertion
Ref Expression
dvdsrval = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
Distinct variable groups:   𝑥,𝑦,   𝑥,𝑧,𝐵,𝑦   𝑥,𝑅,𝑦,𝑧   𝑥, · ,𝑦,𝑧
Allowed substitution hint:   (𝑧)

Proof of Theorem dvdsrval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dvdsr.2 . . 3 = (∥r𝑅)
2 fveq2 6717 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvdsr.1 . . . . . . . . 9 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2796 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54eleq2d 2823 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥𝐵))
64rexeqdv 3326 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦))
75, 6anbi12d 634 . . . . . 6 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦)))
8 fveq2 6717 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvdsr.3 . . . . . . . . . . 11 · = (.r𝑅)
108, 9eqtr4di 2796 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = · )
1110oveqd 7230 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(.r𝑟)𝑥) = (𝑧 · 𝑥))
1211eqeq1d 2739 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(.r𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
1312rexbidv 3216 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
1413anbi2d 632 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
157, 14bitrd 282 . . . . 5 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
1615opabbidv 5119 . . . 4 (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
17 df-dvdsr 19659 . . . 4 r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)})
183fvexi 6731 . . . . 5 𝐵 ∈ V
19 eqcom 2744 . . . . . . . . 9 ((𝑧 · 𝑥) = 𝑦𝑦 = (𝑧 · 𝑥))
2019rexbii 3170 . . . . . . . 8 (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥))
2120abbii 2808 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)}
2218abrexex 7735 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
2321, 22eqeltri 2834 . . . . . 6 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
2423a1i 11 . . . . 5 (𝑥𝐵 → {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
2518, 24opabex3 7740 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
2616, 17, 25fvmpt 6818 . . 3 (𝑅 ∈ V → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
271, 26syl5eq 2790 . 2 (𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
28 fvprc 6709 . . . 4 𝑅 ∈ V → (∥r𝑅) = ∅)
291, 28syl5eq 2790 . . 3 𝑅 ∈ V → = ∅)
30 opabn0 5434 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
31 n0i 4248 . . . . . . . 8 (𝑥𝐵 → ¬ 𝐵 = ∅)
32 fvprc 6709 . . . . . . . . 9 𝑅 ∈ V → (Base‘𝑅) = ∅)
333, 32syl5eq 2790 . . . . . . . 8 𝑅 ∈ V → 𝐵 = ∅)
3431, 33nsyl2 143 . . . . . . 7 (𝑥𝐵𝑅 ∈ V)
3534adantr 484 . . . . . 6 ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3635exlimivv 1940 . . . . 5 (∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3730, 36sylbi 220 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
3837necon1bi 2969 . . 3 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
3929, 38eqtr4d 2780 . 2 𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
4027, 39pm2.61i 185 1 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wne 2940  wrex 3062  Vcvv 3408  c0 4237  {copab 5115  cfv 6380  (class class class)co 7213  Basecbs 16760  .rcmulr 16803  rcdsr 19656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-dvdsr 19659
This theorem is referenced by:  dvdsr  19664  dvdsrpropd  19714  dvdsrzring  20448
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