Theorem eldiophss 43053

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 43044	. 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 3443	. . . . . . . 8 ⊢ 𝑑 ∈ V
4		eqeq1 2739	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
5	4	anbi1d 632	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)))
6	5	rexbidv 3159	. . . . . . . 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 3633	. . . . . . 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 13471	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
10	9	ssriv 3936	. . . . . . . . . . . . 13 ⊢ (1...𝐵) ⊆ ℕ
11		elmapssres 8806	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ (ℕ0 ↑m ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
12	10, 11	mpan2 692	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℕ0 ↑m ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
13	12	ad2antlr 728	. . . . . . . . . . 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 2835	. . . . . . . . . 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 3136	. . . . . . 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 3938	. . . . 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 3967	. . 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 3139	. 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 1542   ∈ wcel 2114  {cab 2713  ∃wrex 3059   ⊆ wss 3900   ↾ cres 5625  ‘cfv 6491  (class class class)co 7358   ↑_m cmap 8765  0cc0 11028  1c1 11029  ℕcn 12147  ℕ₀cn0 12403  ...cfz 13425  mzPolycmzp 43001  Diophcdioph 43034

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-hash 14256  df-mzpcl 43002  df-mzp 43003  df-dioph 43035

This theorem is referenced by: (None)