Theorem eldiophss 42730

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 ⊆ (ℕ0 ↑m (1...𝐵)))

Proof of Theorem eldiophss

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldioph3b 42721	. 2 ⊢ (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}))
2		simpr 484	. . . 4 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)})
3		vex 3492	. . . . . . . 8 ⊢ 𝑑 ∈ V
4		eqeq1 2744	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
5	4	anbi1d 630	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)))
6	5	rexbidv 3185	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)))
7	3, 6	elab 3694	. . . . . . 7 ⊢ (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0))
8		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9		elfznn 13613	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
10	9	ssriv 4012	. . . . . . . . . . . . 13 ⊢ (1...𝐵) ⊆ ℕ
11		elmapssres 8925	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ (ℕ0 ↑m ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
12	10, 11	mpan2 690	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℕ0 ↑m ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
13	12	ad2antlr 726	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
14	8, 13	eqeltrd 2844	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵)))
15	14	ex 412	. . . . . . . . 9 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵))))
16	15	adantrd 491	. . . . . . . 8 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵))))
17	16	rexlimdva 3161	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) → (∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵))))
18	7, 17	biimtrid 242	. . . . . 6 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) → (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} → 𝑑 ∈ (ℕ0 ↑m (1...𝐵))))
19	18	ssrdv 4014	. . . . 5 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} ⊆ (ℕ0 ↑m (1...𝐵)))
20	19	adantr 480	. . . 4 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} ⊆ (ℕ0 ↑m (1...𝐵)))
21	2, 20	eqsstrd 4047	. . 3 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵)))
22	21	r19.29an 3164	. 2 ⊢ ((𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵)))
23	1, 22	sylbi 217	1 ⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ⊆ wss 3976   ↾ cres 5702  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  0cc0 11184  1c1 11185  ℕcn 12293  ℕ₀cn0 12553  ...cfz 13567  mzPolycmzp 42678  Diophcdioph 42711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380  df-mzpcl 42679  df-mzp 42680  df-dioph 42712

This theorem is referenced by: (None)