Theorem inftmrel 33160

Description: The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypothesis

Ref	Expression
inftm.b	⊢ 𝐵 = (Base‘𝑊)

Assertion

Ref	Expression
inftmrel	⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))

Proof of Theorem inftmrel

Dummy variables 𝑥 𝑤 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3509	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		fveq2 6920	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		inftm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
5	4	eleq2d 2830	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ 𝐵))
6	4	eleq2d 2830	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦 ∈ 𝐵))
7	5, 6	anbi12d 631	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
8		fveq2 6920	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
9		fveq2 6920	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10		eqidd 2741	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
11	8, 9, 10	breq123d 5180	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((0g‘𝑤)(lt‘𝑤)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑥))
12		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = (.g‘𝑊))
13	12	oveqd 7465	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑛(.g‘𝑤)𝑥) = (𝑛(.g‘𝑊)𝑥))
14		eqidd 2741	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
15	13, 9, 14	breq123d 5180	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))
16	15	ralbidv 3184	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))
17	11, 16	anbi12d 631	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦)))
18	7, 17	anbi12d 631	. . . . 5 ⊢ (𝑤 = 𝑊 → (((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))))
19	18	opabbidv 5232	. . . 4 ⊢ (𝑤 = 𝑊 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))})
20		df-inftm 33158	. . . 4 ⊢ ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))})
21	3	fvexi 6934	. . . . . 6 ⊢ 𝐵 ∈ V
22	21, 21	xpex 7788	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
23		opabssxp 5792	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))} ⊆ (𝐵 × 𝐵)
24	22, 23	ssexi 5340	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))} ∈ V
25	19, 20, 24	fvmpt 7029	. . 3 ⊢ (𝑊 ∈ V → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))})
26	1, 25	syl 17	. 2 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))})
27	26, 23	eqsstrdi 4063	1 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976   class class class wbr 5166  {copab 5228   × cxp 5698  ‘cfv 6573  (class class class)co 7448  ℕcn 12293  Basecbs 17258  0_gc0g 17499  ltcplt 18378  ._gcmg 19107  ⋘cinftm 33156

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-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  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-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-inftm 33158

This theorem is referenced by:  isarchi  33162