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Theorem inftmrel 30250
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 3400 . . 3 (𝑊𝑉𝑊 ∈ V)
2 fveq2 6411 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 inftm.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
42, 3syl6eqr 2851 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
54eleq2d 2864 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
64eleq2d 2864 . . . . . . 7 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
75, 6anbi12d 625 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
8 fveq2 6411 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
9 fveq2 6411 . . . . . . . 8 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10 eqidd 2800 . . . . . . . 8 (𝑤 = 𝑊𝑥 = 𝑥)
118, 9, 10breq123d 4857 . . . . . . 7 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑥))
12 fveq2 6411 . . . . . . . . . 10 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
1312oveqd 6895 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛(.g𝑊)𝑥))
14 eqidd 2800 . . . . . . . . 9 (𝑤 = 𝑊𝑦 = 𝑦)
1513, 9, 14breq123d 4857 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1615ralbidv 3167 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1711, 16anbi12d 625 . . . . . 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 625 . . . . 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 4909 . . . 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 30248 . . . 4 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
213fvexi 6425 . . . . . 6 𝐵 ∈ V
2221, 21xpex 7196 . . . . 5 (𝐵 × 𝐵) ∈ V
23 opabssxp 5398 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ⊆ (𝐵 × 𝐵)
2422, 23ssexi 4998 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ∈ V
2519, 20, 24fvmpt 6507 . . 3 (𝑊 ∈ V → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))})
261, 25syl 17 . 2 (𝑊𝑉 → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))})
2726, 23syl6eqss 3851 1 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  wss 3769   class class class wbr 4843  {copab 4905   × cxp 5310  cfv 6101  (class class class)co 6878  cn 11312  Basecbs 16184  0gc0g 16415  ltcplt 17256  .gcmg 17856  cinftm 30246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-inftm 30248
This theorem is referenced by:  isarchi  30252
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