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Theorem inftmrel 32596
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 3491 . . 3 (𝑊𝑉𝑊 ∈ V)
2 fveq2 6890 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 inftm.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
42, 3eqtr4di 2788 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
54eleq2d 2817 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
64eleq2d 2817 . . . . . . 7 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
75, 6anbi12d 629 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
8 fveq2 6890 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
9 fveq2 6890 . . . . . . . 8 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10 eqidd 2731 . . . . . . . 8 (𝑤 = 𝑊𝑥 = 𝑥)
118, 9, 10breq123d 5161 . . . . . . 7 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑥))
12 fveq2 6890 . . . . . . . . . 10 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
1312oveqd 7428 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛(.g𝑊)𝑥))
14 eqidd 2731 . . . . . . . . 9 (𝑤 = 𝑊𝑦 = 𝑦)
1513, 9, 14breq123d 5161 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1615ralbidv 3175 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1711, 16anbi12d 629 . . . . . 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 629 . . . . 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 5213 . . . 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 32594 . . . 4 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
213fvexi 6904 . . . . . 6 𝐵 ∈ V
2221, 21xpex 7742 . . . . 5 (𝐵 × 𝐵) ∈ V
23 opabssxp 5767 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ⊆ (𝐵 × 𝐵)
2422, 23ssexi 5321 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ∈ V
2519, 20, 24fvmpt 6997 . . 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, 23eqsstrdi 4035 1 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  Vcvv 3472  wss 3947   class class class wbr 5147  {copab 5209   × cxp 5673  cfv 6542  (class class class)co 7411  cn 12216  Basecbs 17148  0gc0g 17389  ltcplt 18265  .gcmg 18986  cinftm 32592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-inftm 32594
This theorem is referenced by:  isarchi  32598
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