Theorem ex-opab 31090

Description: Example for df-opab 5174. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Assertion

Ref	Expression
ex-opab	⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem ex-opab

Step	Hyp	Ref	Expression
1		3cn 12464	. . 3 ⊢ 3 ∈ ℂ
2		4cn 12468	. . 3 ⊢ 4 ∈ ℂ
3		3p1e4 12527	. . 3 ⊢ (3 + 1) = 4
4	1	elexi 3476	. . . 4 ⊢ 3 ∈ V
5	2	elexi 3476	. . . 4 ⊢ 4 ∈ V
6		eleq1 2852	. . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ))
7		oveq1 7458	. . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1))
8	7	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦))
9	6, 8	3anbi13d 1467	. . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦)))
10		eleq1 2852	. . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ))
11		eqeq2 2775	. . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4))
12	10, 11	3anbi23d 1468	. . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)))
13		eqid 2763	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}
14	4, 5, 9, 12, 13	brab 5525	. . 3 ⊢ (3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))
15	1, 2, 3, 14	mpbir3an 1360	. 2 ⊢ 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4
16		breq 5110	. 2 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → (3𝑅4 ↔ 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4))
17	15, 16	mpbiri 261	1 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5108  {copab 5173  (class class class)co 7451  ℂcc 11237  1c1 11240   + caddc 11242  3c3 12438  4c4 12439

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 5255  ax-pr 5398  ax-1cn 11297  ax-addcl 11299

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 4841  df-uni 4873  df-br 5109  df-opab 5174  df-iota 6520  df-fv 6574  df-ov 7454  df-2 12445  df-3 12446  df-4 12447

This theorem is used by: (None)