Theorem inftmrel 32016

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 3463	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		fveq2 6842	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		inftm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
5	4	eleq2d 2823	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ 𝐵))
6	4	eleq2d 2823	. . . . . . 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 6842	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
9		fveq2 6842	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10		eqidd 2737	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
11	8, 9, 10	breq123d 5119	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((0g‘𝑤)(lt‘𝑤)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑥))
12		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = (.g‘𝑊))
13	12	oveqd 7374	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑛(.g‘𝑤)𝑥) = (𝑛(.g‘𝑊)𝑥))
14		eqidd 2737	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
15	13, 9, 14	breq123d 5119	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g‘𝑊)𝑥)(lt‘𝑊)𝑦))
16	15	ralbidv 3174	. . . . . . 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 5171	. . . 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 32014	. . . 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 6856	. . . . . 6 ⊢ 𝐵 ∈ V
22	21, 21	xpex 7687	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
23		opabssxp 5724	. . . . 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 5279	. . . 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 6948	. . 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 3998	1 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ⊆ wss 3910   class class class wbr 5105  {copab 5167   × cxp 5631  ‘cfv 6496  (class class class)co 7357  ℕcn 12153  Basecbs 17083  0_gc0g 17321  ltcplt 18197  ._gcmg 18872  ⋘cinftm 32012

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-inftm 32014

This theorem is referenced by:  isarchi  32018