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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heiborlem2.5	⊢ 𝐴 ∈ V
heiborlem2.6	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
heiborlem2	⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (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

Step	Hyp	Ref	Expression
1		heiborlem2.5	. 2 ⊢ 𝐴 ∈ V
2		heiborlem2.6	. 2 ⊢ 𝐶 ∈ V
3		eleq1 2822	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝑛)))
4		oveq1 7411	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
5	4	eleq1d 2819	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
6	3, 5	3anbi23d 1440	. 2 ⊢ (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7		eleq1 2822	. . 3 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ ℕ0 ↔ 𝐶 ∈ ℕ0))
8		fveq2 6888	. . . 4 ⊢ (𝑛 = 𝐶 → (𝐹‘𝑛) = (𝐹‘𝐶))
9	8	eleq2d 2820	. . 3 ⊢ (𝑛 = 𝐶 → (𝐴 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝐶)))
10		oveq2 7412	. . . 4 ⊢ (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
11	10	eleq1d 2819	. . 3 ⊢ (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
12	7, 9, 11	3anbi123d 1437	. 2 ⊢ (𝑛 = 𝐶 → ((𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾) ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)))
13		heibor.4	. 2 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
14	1, 2, 6, 12, 13	brab 5542	1 ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))

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  ℕ₀cn0 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