Theorem heiborlem2 37772

Description: Lemma for heibor 37781. 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 2832	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝑛)))
4		oveq1 7455	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
5	4	eleq1d 2829	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
6	3, 5	3anbi23d 1439	. 2 ⊢ (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7		eleq1 2832	. . 3 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ ℕ0 ↔ 𝐶 ∈ ℕ0))
8		fveq2 6920	. . . 4 ⊢ (𝑛 = 𝐶 → (𝐹‘𝑛) = (𝐹‘𝐶))
9	8	eleq2d 2830	. . 3 ⊢ (𝑛 = 𝐶 → (𝐴 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝐶)))
10		oveq2 7456	. . . 4 ⊢ (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
11	10	eleq1d 2829	. . 3 ⊢ (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
12	7, 9, 11	3anbi123d 1436	. 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 5562	1 ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  {copab 5228  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℕ₀cn0 12553  MetOpencmopn 21377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  heiborlem3  37773  heiborlem5  37775  heiborlem6  37776  heiborlem8  37778  heiborlem10  37780