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Theorem heiborlem2 34090
Description: Lemma for heibor 34099. 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 2864 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝑛)))
4 oveq1 6883 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
54eleq1d 2861 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
63, 53anbi23d 1564 . 2 (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0𝐴 ∈ (𝐹𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7 eleq1 2864 . . 3 (𝑛 = 𝐶 → (𝑛 ∈ ℕ0𝐶 ∈ ℕ0))
8 fveq2 6409 . . . 4 (𝑛 = 𝐶 → (𝐹𝑛) = (𝐹𝐶))
98eleq2d 2862 . . 3 (𝑛 = 𝐶 → (𝐴 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝐶)))
10 oveq2 6884 . . . 4 (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
1110eleq1d 2861 . . 3 (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
127, 9, 113anbi123d 1561 . 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 5192 1 (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  w3a 1108   = wceq 1653  wcel 2157  {cab 2783  wrex 3088  Vcvv 3383  cin 3766  wss 3767  𝒫 cpw 4347   cuni 4626   class class class wbr 4841  {copab 4903  cfv 6099  (class class class)co 6876  Fincfn 8193  0cn0 11576  MetOpencmopn 20055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-iota 6062  df-fv 6107  df-ov 6879
This theorem is referenced by:  heiborlem3  34091  heiborlem5  34093  heiborlem6  34094  heiborlem8  34096  heiborlem10  34098
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