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Theorem heiborlem2 34971
Description: Lemma for heibor 34980. 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 2897 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝑛)))
4 oveq1 7152 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
54eleq1d 2894 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
63, 53anbi23d 1430 . 2 (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0𝐴 ∈ (𝐹𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7 eleq1 2897 . . 3 (𝑛 = 𝐶 → (𝑛 ∈ ℕ0𝐶 ∈ ℕ0))
8 fveq2 6663 . . . 4 (𝑛 = 𝐶 → (𝐹𝑛) = (𝐹𝐶))
98eleq2d 2895 . . 3 (𝑛 = 𝐶 → (𝐴 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝐶)))
10 oveq2 7153 . . . 4 (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
1110eleq1d 2894 . . 3 (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
127, 9, 113anbi123d 1427 . 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 5421 1 (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  {copab 5119  cfv 6348  (class class class)co 7145  Fincfn 8497  0cn0 11885  MetOpencmopn 20463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  heiborlem3  34972  heiborlem5  34974  heiborlem6  34975  heiborlem8  34977  heiborlem10  34979
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