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Theorem ex-opab 30262
Description: Example for df-opab 5215. 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
StepHypRef Expression
1 3cn 12331 . . 3 3 ∈ ℂ
2 4cn 12335 . . 3 4 ∈ ℂ
3 3p1e4 12395 . . 3 (3 + 1) = 4
41elexi 3493 . . . 4 3 ∈ V
52elexi 3493 . . . 4 4 ∈ V
6 eleq1 2817 . . . . 5 (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ))
7 oveq1 7433 . . . . . 6 (𝑥 = 3 → (𝑥 + 1) = (3 + 1))
87eqeq1d 2730 . . . . 5 (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦))
96, 83anbi13d 1434 . . . 4 (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦)))
10 eleq1 2817 . . . . 5 (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ))
11 eqeq2 2740 . . . . 5 (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4))
1210, 113anbi23d 1435 . . . 4 (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)))
13 eqid 2728 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}
144, 5, 9, 12, 13brab 5549 . . 3 (3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))
151, 2, 3, 14mpbir3an 1338 . 2 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4
16 breq 5154 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → (3𝑅4 ↔ 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4))
1715, 16mpbiri 257 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5152  {copab 5214  (class class class)co 7426  cc 11144  1c1 11147   + caddc 11149  3c3 12306  4c4 12307
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-1cn 11204  ax-addcl 11206
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-iota 6505  df-fv 6561  df-ov 7429  df-2 12313  df-3 12314  df-4 12315
This theorem is referenced by: (None)
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