Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  heiborlem2 Structured version   Visualization version   GIF version

Theorem heiborlem2 37819
Description: Lemma for heibor 37828. 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 2829 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝑛)))
4 oveq1 7438 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
54eleq1d 2826 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
63, 53anbi23d 1441 . 2 (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0𝐴 ∈ (𝐹𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7 eleq1 2829 . . 3 (𝑛 = 𝐶 → (𝑛 ∈ ℕ0𝐶 ∈ ℕ0))
8 fveq2 6906 . . . 4 (𝑛 = 𝐶 → (𝐹𝑛) = (𝐹𝐶))
98eleq2d 2827 . . 3 (𝑛 = 𝐶 → (𝐴 ∈ (𝐹𝑛) ↔ 𝐴 ∈ (𝐹𝐶)))
10 oveq2 7439 . . . 4 (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
1110eleq1d 2826 . . 3 (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
127, 9, 113anbi123d 1438 . 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 5548 1 (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  {copab 5205  cfv 6561  (class class class)co 7431  Fincfn 8985  0cn0 12526  MetOpencmopn 21354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  heiborlem3  37820  heiborlem5  37822  heiborlem6  37823  heiborlem8  37825  heiborlem10  37827
  Copyright terms: Public domain W3C validator