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Theorem dvdsrval 20361
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 6906 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvdsr.1 . . . . . . . . 9 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2795 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54eleq2d 2827 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥𝐵))
64rexeqdv 3327 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦))
75, 6anbi12d 632 . . . . . 6 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦)))
8 fveq2 6906 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvdsr.3 . . . . . . . . . . 11 · = (.r𝑅)
108, 9eqtr4di 2795 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = · )
1110oveqd 7448 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(.r𝑟)𝑥) = (𝑧 · 𝑥))
1211eqeq1d 2739 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(.r𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
1312rexbidv 3179 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
1413anbi2d 630 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
157, 14bitrd 279 . . . . 5 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
1615opabbidv 5209 . . . 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 20357 . . . 4 r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)})
183fvexi 6920 . . . . 5 𝐵 ∈ V
19 eqcom 2744 . . . . . . . . 9 ((𝑧 · 𝑥) = 𝑦𝑦 = (𝑧 · 𝑥))
2019rexbii 3094 . . . . . . . 8 (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥))
2120abbii 2809 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)}
2218abrexex 7987 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
2321, 22eqeltri 2837 . . . . . 6 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
2423a1i 11 . . . . 5 (𝑥𝐵 → {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
2518, 24opabex3 7992 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
2616, 17, 25fvmpt 7016 . . 3 (𝑅 ∈ V → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
271, 26eqtrid 2789 . 2 (𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
28 fvprc 6898 . . . 4 𝑅 ∈ V → (∥r𝑅) = ∅)
291, 28eqtrid 2789 . . 3 𝑅 ∈ V → = ∅)
30 opabn0 5558 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
31 n0i 4340 . . . . . . . 8 (𝑥𝐵 → ¬ 𝐵 = ∅)
32 fvprc 6898 . . . . . . . . 9 𝑅 ∈ V → (Base‘𝑅) = ∅)
333, 32eqtrid 2789 . . . . . . . 8 𝑅 ∈ V → 𝐵 = ∅)
3431, 33nsyl2 141 . . . . . . 7 (𝑥𝐵𝑅 ∈ V)
3534adantr 480 . . . . . 6 ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3635exlimivv 1932 . . . . 5 (∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3730, 36sylbi 217 . . . 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 182 1 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wrex 3070  Vcvv 3480  c0 4333  {copab 5205  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  rcdsr 20354
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-dvdsr 20357
This theorem is referenced by:  dvdsr  20362  dvdsrpropd  20416  dvdsrzring  21472
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