Theorem eldiophss 43238

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 43229	. 2 ⊢ (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}))
2		simpr 486	. . . 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 3437	. . . . . . . 8 ⊢ 𝑑 ∈ V
4		eqeq1 2745	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵))))
5	4	anbi1d 638	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)))
6	5	rexbidv 3165	. . . . . . . 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 3619	. . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵)))
9		elfznn 13502	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ)
10	9	ssriv 3921	. . . . . . . . . . . . 13 ⊢ (1...𝐵) ⊆ ℕ
11		elmapssres 8808	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ (ℕ0 ↑m ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
12	10, 11	mpan2 698	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℕ0 ↑m ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵)))
13	12	ad2antlr 734	. . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵)))
15	14	ex 414	. . . . . . . . 9 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵))))
16	15	adantrd 493	. . . . . . . 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 3142	. . . . . . 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 244	. . . . . 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 3923	. . . . 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 482	. . . 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 3951	. . 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 3145	. 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 219	1 ⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∃wrex 3065   ⊆ wss 3885   ↾ cres 5623  ‘cfv 6489  (class class class)co 7360   ↑_m cmap 8767  0cc0 11033  1c1 11034  ℕcn 12169  ℕ₀cn0 12432  ...cfz 13456  mzPolycmzp 43186  Diophcdioph 43219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-hash 14288  df-mzpcl 43187  df-mzp 43188  df-dioph 43220

This theorem is referenced by: (None)