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Theorem eldiophss 40133
 Description: Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Assertion
Ref Expression
eldiophss (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0m (1...𝐵)))

Proof of Theorem eldiophss
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldioph3b 40124 . 2 (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}))
2 simpr 488 . . . 4 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)})
3 vex 3413 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2762 . . . . . . . . . 10 (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
54anbi1d 632 . . . . . . . . 9 (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
65rexbidv 3221 . . . . . . . 8 (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
73, 6elab 3590 . . . . . . 7 (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0))
8 simpr 488 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9 elfznn 12998 . . . . . . . . . . . . . 14 (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
109ssriv 3898 . . . . . . . . . . . . 13 (1...𝐵) ⊆ ℕ
11 elmapssres 8462 . . . . . . . . . . . . 13 ((𝑐 ∈ (ℕ0m ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0m (1...𝐵)))
1210, 11mpan2 690 . . . . . . . . . . . 12 (𝑐 ∈ (ℕ0m ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0m (1...𝐵)))
1312ad2antlr 726 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0m (1...𝐵)))
148, 13eqeltrd 2852 . . . . . . . . . 10 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0m (1...𝐵)))
1514ex 416 . . . . . . . . 9 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0m (1...𝐵))))
1615adantrd 495 . . . . . . . 8 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0m ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0m (1...𝐵))))
1716rexlimdva 3208 . . . . . . 7 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (∃𝑐 ∈ (ℕ0m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0m (1...𝐵))))
187, 17syl5bi 245 . . . . . 6 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} → 𝑑 ∈ (ℕ0m (1...𝐵))))
1918ssrdv 3900 . . . . 5 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ⊆ (ℕ0m (1...𝐵)))
2019adantr 484 . . . 4 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ⊆ (ℕ0m (1...𝐵)))
212, 20eqsstrd 3932 . . 3 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0m (1...𝐵)))
2221r19.29an 3212 . 2 ((𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0m (1...𝐵)))
231, 22sylbi 220 1 (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0m (1...𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ⊆ wss 3860   ↾ cres 5530  ‘cfv 6340  (class class class)co 7156   ↑m cmap 8422  0cc0 10588  1c1 10589  ℕcn 11687  ℕ0cn0 11947  ...cfz 12952  mzPolycmzp 40081  Diophcdioph 40114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-inf2 9150  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-oadd 8122  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-dju 9376  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-n0 11948  df-z 12034  df-uz 12296  df-fz 12953  df-hash 13754  df-mzpcl 40082  df-mzp 40083  df-dioph 40115 This theorem is referenced by: (None)
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