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Theorem inftmrel 30068
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 3361 . . 3 (𝑊𝑉𝑊 ∈ V)
2 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 inftm.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
42, 3syl6eqr 2822 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
54eleq2d 2835 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
64eleq2d 2835 . . . . . . 7 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
75, 6anbi12d 608 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
8 fveq2 6332 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
9 fveq2 6332 . . . . . . . 8 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10 eqidd 2771 . . . . . . . 8 (𝑤 = 𝑊𝑥 = 𝑥)
118, 9, 10breq123d 4798 . . . . . . 7 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑥))
12 fveq2 6332 . . . . . . . . . 10 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
1312oveqd 6809 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛(.g𝑊)𝑥))
14 eqidd 2771 . . . . . . . . 9 (𝑤 = 𝑊𝑦 = 𝑦)
1513, 9, 14breq123d 4798 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1615ralbidv 3134 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1711, 16anbi12d 608 . . . . . 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 608 . . . . 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 4848 . . . 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 30066 . . . 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 6342 . . . . . . 7 (Base‘𝑊) ∈ V
223, 21eqeltri 2845 . . . . . 6 𝐵 ∈ V
2322, 22xpex 7108 . . . . 5 (𝐵 × 𝐵) ∈ V
24 opabssxp 5333 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ⊆ (𝐵 × 𝐵)
2523, 24ssexi 4934 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ∈ V
2619, 20, 25fvmpt 6424 . . 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 3802 1 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  wss 3721   class class class wbr 4784  {copab 4844   × cxp 5247  cfv 6031  (class class class)co 6792  cn 11221  Basecbs 16063  0gc0g 16307  ltcplt 17148  .gcmg 17747  cinftm 30064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-inftm 30066
This theorem is referenced by:  isarchi  30070
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