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Theorem inftmrel 30043
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 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))

Proof of Theorem inftmrel
Dummy variables 𝑥 𝑤 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3352 . . 3 (𝑊𝑉𝑊 ∈ V)
2 fveq2 6352 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 inftm.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
42, 3syl6eqr 2812 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
54eleq2d 2825 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
64eleq2d 2825 . . . . . . 7 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
75, 6anbi12d 749 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
8 fveq2 6352 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
9 fveq2 6352 . . . . . . . 8 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10 eqidd 2761 . . . . . . . 8 (𝑤 = 𝑊𝑥 = 𝑥)
118, 9, 10breq123d 4818 . . . . . . 7 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑥))
12 fveq2 6352 . . . . . . . . . 10 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
1312oveqd 6830 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛(.g𝑊)𝑥))
14 eqidd 2761 . . . . . . . . 9 (𝑤 = 𝑊𝑦 = 𝑦)
1513, 9, 14breq123d 4818 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1615ralbidv 3124 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1711, 16anbi12d 749 . . . . . 6 (𝑤 = 𝑊 → (((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦)))
187, 17anbi12d 749 . . . . 5 (𝑤 = 𝑊 → (((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))))
1918opabbidv 4868 . . . 4 (𝑤 = 𝑊 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))})
20 df-inftm 30041 . . . 4 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
21 fvex 6362 . . . . . . 7 (Base‘𝑊) ∈ V
223, 21eqeltri 2835 . . . . . 6 𝐵 ∈ V
2322, 22xpex 7127 . . . . 5 (𝐵 × 𝐵) ∈ V
24 opabssxp 5350 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ⊆ (𝐵 × 𝐵)
2523, 24ssexi 4955 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ∈ V
2619, 20, 25fvmpt 6444 . . 3 (𝑊 ∈ V → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))})
271, 26syl 17 . 2 (𝑊𝑉 → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))})
2827, 24syl6eqss 3796 1 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715   class class class wbr 4804  {copab 4864   × cxp 5264  cfv 6049  (class class class)co 6813  cn 11212  Basecbs 16059  0gc0g 16302  ltcplt 17142  .gcmg 17741  cinftm 30039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-inftm 30041
This theorem is referenced by:  isarchi  30045
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