Theorem ex-opab 31015

Description: Example for df-opab 5168. 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 12405	. . 3 ⊢ 3 ∈ ℂ
2		4cn 12409	. . 3 ⊢ 4 ∈ ℂ
3		3p1e4 12468	. . 3 ⊢ (3 + 1) = 4
4	1	elexi 3473	. . . 4 ⊢ 3 ∈ V
5	2	elexi 3473	. . . 4 ⊢ 4 ∈ V
6		eleq1 2849	. . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ))
7		oveq1 7419	. . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1))
8	7	eqeq1d 2763	. . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦))
9	6, 8	3anbi13d 1466	. . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦)))
10		eleq1 2849	. . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ))
11		eqeq2 2773	. . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4))
12	10, 11	3anbi23d 1467	. . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)))
13		eqid 2761	. . . 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 5518	. . 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 5105	. 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 2145   class class class wbr 5103  {copab 5167  (class class class)co 7412  ℂcc 11179  1c1 11182   + caddc 11184  3c3 12379  4c4 12380

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-1cn 11239  ax-addcl 11241

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6487  df-fv 6539  df-ov 7415  df-2 12386  df-3 12387  df-4 12388

This theorem is used by: (None)