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Theorem heiborlem2 32781
Description: Lemma for heibor 32790. Substitutions for the set 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor.3 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
heibor.4 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heiborlem2.5 𝐴 ∈ V
heiborlem2.6 𝐶 ∈ V
Assertion
Ref Expression
heiborlem2 (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Distinct variable groups:   𝑦,𝑛,𝐴   𝑢,𝑛,𝐹,𝑦   𝑣,𝑛,𝐷,𝑢,𝑦   𝐵,𝑛,𝑢,𝑣,𝑦   𝑛,𝐽,𝑢,𝑣,𝑦   𝑈,𝑛,𝑢,𝑣,𝑦   𝐶,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑦
Allowed substitution hints:   𝐴(𝑣,𝑢)   𝐹(𝑣)   𝐺(𝑦,𝑣,𝑢,𝑛)   𝐾(𝑣,𝑢)

Proof of Theorem heiborlem2
StepHypRef Expression
1 heiborlem2.5 . 2 𝐴 ∈ V
2 heiborlem2.6 . 2 𝐶 ∈ V
3 eleq1 2676 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝑛)))
4 oveq1 6556 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
54eleq1d 2672 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
63, 53anbi23d 1394 . 2 (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0𝐴 ∈ (𝐹𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7 eleq1 2676 . . 3 (𝑛 = 𝐶 → (𝑛 ∈ ℕ0𝐶 ∈ ℕ0))
8 fveq2 6103 . . . 4 (𝑛 = 𝐶 → (𝐹𝑛) = (𝐹𝐶))
98eleq2d 2673 . . 3 (𝑛 = 𝐶 → (𝐴 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝐶)))
10 oveq2 6557 . . . 4 (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
1110eleq1d 2672 . . 3 (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
127, 9, 113anbi123d 1391 . 2 (𝑛 = 𝐶 → ((𝑛 ∈ ℕ0𝐴 ∈ (𝐹𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾) ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)))
13 heibor.4 . 2 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
141, 2, 6, 12, 13brab 4923 1 (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  w3a 1031   = wceq 1475  wcel 1977  {cab 2596  wrex 2897  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372   class class class wbr 4583  {copab 4642  cfv 5804  (class class class)co 6549  Fincfn 7841  0cn0 11169  MetOpencmopn 19557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  heiborlem3  32782  heiborlem5  32784  heiborlem6  32785  heiborlem8  32787  heiborlem10  32789
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