Theorem ex-opab 30189

Description: Example for df-opab 5204. 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 12294	. . 3 ⊢ 3 ∈ ℂ
2		4cn 12298	. . 3 ⊢ 4 ∈ ℂ
3		3p1e4 12358	. . 3 ⊢ (3 + 1) = 4
4	1	elexi 3488	. . . 4 ⊢ 3 ∈ V
5	2	elexi 3488	. . . 4 ⊢ 4 ∈ V
6		eleq1 2815	. . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ))
7		oveq1 7411	. . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1))
8	7	eqeq1d 2728	. . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦))
9	6, 8	3anbi13d 1434	. . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦)))
10		eleq1 2815	. . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ))
11		eqeq2 2738	. . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4))
12	10, 11	3anbi23d 1435	. . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)))
13		eqid 2726	. . . 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 5536	. . 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 1338	. 2 ⊢ 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4
16		breq 5143	. 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 258	1 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  {copab 5203  (class class class)co 7404  ℂcc 11107  1c1 11110   + caddc 11112  3c3 12269  4c4 12270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-1cn 11167  ax-addcl 11169

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-iota 6488  df-fv 6544  df-ov 7407  df-2 12276  df-3 12277  df-4 12278

This theorem is referenced by: (None)