Theorem inftmrel 33321

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 3474	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		fveq2 6862	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		inftm.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
5	4	eleq2d 2847	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ 𝐵))
6	4	eleq2d 2847	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦 ∈ 𝐵))
7	5, 6	anbi12d 641	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
8		fveq2 6862	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
9		fveq2 6862	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10		eqidd 2762	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
11	8, 9, 10	breq123d 5111	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((0g‘𝑤)(lt‘𝑤)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑥))
12		fveq2 6862	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = (.g‘𝑊))
13	12	oveqd 7408	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑛(.g‘𝑤)𝑥) = (𝑛(.g‘𝑊)𝑥))
14		eqidd 2762	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
15	13, 9, 14	breq123d 5111	. . . . . . . 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 641	. . . . . 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 641	. . . . 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 5163	. . . 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 33319	. . . 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 6876	. . . . . 6 ⊢ 𝐵 ∈ V
22	21, 21	xpex 7731	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
23		opabssxp 5735	. . . . 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 5275	. . . 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 6970	. . 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 3978	1 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3902   class class class wbr 5097  {copab 5159   × cxp 5641  ‘cfv 6516  (class class class)co 7391  ℕcn 12204  Basecbs 17236  0_gc0g 17459  ltcplt 18331  ._gcmg 19100  ⋘cinftm 33317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-inftm 33319

This theorem is referenced by:  isarchi  33323