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Theorem inftmrel 30804
 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 3512 . . 3 (𝑊𝑉𝑊 ∈ V)
2 fveq2 6664 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3 inftm.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
42, 3syl6eqr 2874 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
54eleq2d 2898 . . . . . . 7 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
64eleq2d 2898 . . . . . . 7 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
75, 6anbi12d 632 . . . . . 6 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
8 fveq2 6664 . . . . . . . 8 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
9 fveq2 6664 . . . . . . . 8 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
10 eqidd 2822 . . . . . . . 8 (𝑤 = 𝑊𝑥 = 𝑥)
118, 9, 10breq123d 5072 . . . . . . 7 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑥))
12 fveq2 6664 . . . . . . . . . 10 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
1312oveqd 7167 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛(.g𝑊)𝑥))
14 eqidd 2822 . . . . . . . . 9 (𝑤 = 𝑊𝑦 = 𝑦)
1513, 9, 14breq123d 5072 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1615ralbidv 3197 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))
1711, 16anbi12d 632 . . . . . 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 632 . . . . 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 5124 . . . 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 30802 . . . 4 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
213fvexi 6678 . . . . . 6 𝐵 ∈ V
2221, 21xpex 7470 . . . . 5 (𝐵 × 𝐵) ∈ V
23 opabssxp 5637 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ⊆ (𝐵 × 𝐵)
2422, 23ssexi 5218 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑊)𝑥)(lt‘𝑊)𝑦))} ∈ V
2519, 20, 24fvmpt 6762 . . 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 4020 1 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ⊆ wss 3935   class class class wbr 5058  {copab 5120   × cxp 5547  ‘cfv 6349  (class class class)co 7150  ℕcn 11632  Basecbs 16477  0gc0g 16707  ltcplt 17545  .gcmg 18218  ⋘cinftm 30800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-inftm 30802 This theorem is referenced by:  isarchi  30806
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