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Theorem dvdsrval 18409
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 6083 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvdsr.1 . . . . . . . . 9 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2656 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54eleq2d 2667 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥𝐵))
64rexeqdv 3116 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦))
75, 6anbi12d 742 . . . . . 6 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦)))
8 fveq2 6083 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvdsr.3 . . . . . . . . . . 11 · = (.r𝑅)
108, 9syl6eqr 2656 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = · )
1110oveqd 6539 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(.r𝑟)𝑥) = (𝑧 · 𝑥))
1211eqeq1d 2606 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(.r𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
1312rexbidv 3028 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
1413anbi2d 735 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
157, 14bitrd 266 . . . . 5 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
1615opabbidv 4637 . . . 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 18405 . . . 4 r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)})
18 fvex 6093 . . . . . 6 (Base‘𝑅) ∈ V
193, 18eqeltri 2678 . . . . 5 𝐵 ∈ V
20 eqcom 2611 . . . . . . . . 9 ((𝑧 · 𝑥) = 𝑦𝑦 = (𝑧 · 𝑥))
2120rexbii 3017 . . . . . . . 8 (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥))
2221abbii 2720 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)}
2319abrexex 7005 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
2422, 23eqeltri 2678 . . . . . 6 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
2524a1i 11 . . . . 5 (𝑥𝐵 → {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
2619, 25opabex3 7010 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
2716, 17, 26fvmpt 6171 . . 3 (𝑅 ∈ V → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
281, 27syl5eq 2650 . 2 (𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
29 fvprc 6077 . . . 4 𝑅 ∈ V → (∥r𝑅) = ∅)
301, 29syl5eq 2650 . . 3 𝑅 ∈ V → = ∅)
31 opabn0 4916 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
32 n0i 3873 . . . . . . . 8 (𝑥𝐵 → ¬ 𝐵 = ∅)
33 fvprc 6077 . . . . . . . . 9 𝑅 ∈ V → (Base‘𝑅) = ∅)
343, 33syl5eq 2650 . . . . . . . 8 𝑅 ∈ V → 𝐵 = ∅)
3532, 34nsyl2 140 . . . . . . 7 (𝑥𝐵𝑅 ∈ V)
3635adantr 479 . . . . . 6 ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3736exlimivv 1845 . . . . 5 (∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3831, 37sylbi 205 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
3938necon1bi 2804 . . 3 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
4030, 39eqtr4d 2641 . 2 𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
4128, 40pm2.61i 174 1 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wex 1694  wcel 1975  {cab 2590  wne 2774  wrex 2891  Vcvv 3167  c0 3868  {copab 4631  cfv 5785  (class class class)co 6522  Basecbs 15636  .rcmulr 15710  rcdsr 18402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-dvdsr 18405
This theorem is referenced by:  dvdsr  18410  dvdsrpropd  18460  dvdsrzring  19591
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