Theorem heiborlem2 38937

Description: Lemma for heibor 38946. 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 2852	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝑛)))
4		oveq1 7447	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛))
5	4	eleq1d 2849	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾))
6	3, 5	3anbi23d 1468	. 2 ⊢ (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾)))
7		eleq1 2852	. . 3 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ ℕ0 ↔ 𝐶 ∈ ℕ0))
8		fveq2 6905	. . . 4 ⊢ (𝑛 = 𝐶 → (𝐹‘𝑛) = (𝐹‘𝐶))
9	8	eleq2d 2850	. . 3 ⊢ (𝑛 = 𝐶 → (𝐴 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝐶)))
10		oveq2 7448	. . . 4 ⊢ (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶))
11	10	eleq1d 2849	. . 3 ⊢ (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾))
12	7, 9, 11	3anbi123d 1465	. 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 5522	1 ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  {copab 5170  ‘cfv 6555  (class class class)co 7440  Fincfn 8998  ℕ₀cn0 12630  MetOpencmopn 21695

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-iota 6509  df-fv 6563  df-ov 7443

This theorem is used by:  heiborlem3  38938  heiborlem5  38940  heiborlem6  38941  heiborlem8  38943  heiborlem10  38945