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Theorem dvdsrval 18853
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 6332 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvdsr.1 . . . . . . . . 9 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2823 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54eleq2d 2836 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥𝐵))
64rexeqdv 3294 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦))
75, 6anbi12d 616 . . . . . 6 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦)))
8 fveq2 6332 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvdsr.3 . . . . . . . . . . 11 · = (.r𝑅)
108, 9syl6eqr 2823 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = · )
1110oveqd 6810 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(.r𝑟)𝑥) = (𝑧 · 𝑥))
1211eqeq1d 2773 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(.r𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
1312rexbidv 3200 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
1413anbi2d 614 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
157, 14bitrd 268 . . . . 5 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
1615opabbidv 4850 . . . 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 18849 . . . 4 r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)})
18 fvex 6342 . . . . . 6 (Base‘𝑅) ∈ V
193, 18eqeltri 2846 . . . . 5 𝐵 ∈ V
20 eqcom 2778 . . . . . . . . 9 ((𝑧 · 𝑥) = 𝑦𝑦 = (𝑧 · 𝑥))
2120rexbii 3189 . . . . . . . 8 (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥))
2221abbii 2888 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)}
2319abrexex 7288 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
2422, 23eqeltri 2846 . . . . . 6 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
2524a1i 11 . . . . 5 (𝑥𝐵 → {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
2619, 25opabex3 7293 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
2716, 17, 26fvmpt 6424 . . 3 (𝑅 ∈ V → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
281, 27syl5eq 2817 . 2 (𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
29 fvprc 6326 . . . 4 𝑅 ∈ V → (∥r𝑅) = ∅)
301, 29syl5eq 2817 . . 3 𝑅 ∈ V → = ∅)
31 opabn0 5139 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
32 n0i 4068 . . . . . . . 8 (𝑥𝐵 → ¬ 𝐵 = ∅)
33 fvprc 6326 . . . . . . . . 9 𝑅 ∈ V → (Base‘𝑅) = ∅)
343, 33syl5eq 2817 . . . . . . . 8 𝑅 ∈ V → 𝐵 = ∅)
3532, 34nsyl2 144 . . . . . . 7 (𝑥𝐵𝑅 ∈ V)
3635adantr 466 . . . . . 6 ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3736exlimivv 2012 . . . . 5 (∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3831, 37sylbi 207 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
3938necon1bi 2971 . . 3 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
4030, 39eqtr4d 2808 . 2 𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
4128, 40pm2.61i 176 1 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wrex 3062  Vcvv 3351  c0 4063  {copab 4846  cfv 6031  (class class class)co 6793  Basecbs 16064  .rcmulr 16150  rcdsr 18846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-dvdsr 18849
This theorem is referenced by:  dvdsr  18854  dvdsrpropd  18904  dvdsrzring  20046
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