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Theorem heiborlem2 36618
Description: Lemma for heibor 36627. 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 2822 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝑛)))
4 oveq1 7411 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
54eleq1d 2819 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
63, 53anbi23d 1440 . 2 (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0𝐴 ∈ (𝐹𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7 eleq1 2822 . . 3 (𝑛 = 𝐶 → (𝑛 ∈ ℕ0𝐶 ∈ ℕ0))
8 fveq2 6888 . . . 4 (𝑛 = 𝐶 → (𝐹𝑛) = (𝐹𝐶))
98eleq2d 2820 . . 3 (𝑛 = 𝐶 → (𝐴 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝐶)))
10 oveq2 7412 . . . 4 (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
1110eleq1d 2819 . . 3 (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
127, 9, 113anbi123d 1437 . 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 5542 1 (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907   class class class wbr 5147  {copab 5209  cfv 6540  (class class class)co 7404  Fincfn 8935  0cn0 12468  MetOpencmopn 20919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-iota 6492  df-fv 6548  df-ov 7407
This theorem is referenced by:  heiborlem3  36619  heiborlem5  36621  heiborlem6  36622  heiborlem8  36624  heiborlem10  36626
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