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Theorem eldiophss 36356
 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 36346 . 2 (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}))
2 simpr 476 . . . 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 3176 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2614 . . . . . . . . . 10 (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
54anbi1d 737 . . . . . . . . 9 (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
65rexbidv 3034 . . . . . . . 8 (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)))
73, 6elab 3319 . . . . . . 7 (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0))
8 simpr 476 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9 elfznn 12241 . . . . . . . . . . . . . 14 (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
109ssriv 3572 . . . . . . . . . . . . 13 (1...𝐵) ⊆ ℕ
11 elmapssres 7768 . . . . . . . . . . . . 13 ((𝑐 ∈ (ℕ0𝑚 ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
1210, 11mpan2 703 . . . . . . . . . . . 12 (𝑐 ∈ (ℕ0𝑚 ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
1312ad2antlr 759 . . . . . . . . . . 11 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0𝑚 (1...𝐵)))
148, 13eqeltrd 2688 . . . . . . . . . 10 ((((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵)))
1514ex 449 . . . . . . . . 9 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1615adantrd 483 . . . . . . . 8 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0𝑚 ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1716rexlimdva 3013 . . . . . . 7 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0) → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
187, 17syl5bi 231 . . . . . 6 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} → 𝑑 ∈ (ℕ0𝑚 (1...𝐵))))
1918ssrdv 3574 . . . . 5 ((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)} ⊆ (ℕ0𝑚 (1...𝐵)))
2019adantr 480 . . . 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 3602 . . 3 (((𝐵 ∈ ℕ0𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
2221r19.29an 3059 . 2 ((𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0𝑚 ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎𝑐) = 0)}) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
231, 22sylbi 206 1 (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0𝑚 (1...𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ⊆ wss 3540   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  0cc0 9815  1c1 9816  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  mzPolycmzp 36303  Diophcdioph 36336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-mzpcl 36304  df-mzp 36305  df-dioph 36337 This theorem is referenced by: (None)
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