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Theorem ex-opab 27143
Description: Example for df-opab 4674. 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 11039 . . 3 3 ∈ ℂ
2 4cn 11042 . . 3 4 ∈ ℂ
3 3p1e4 11097 . . 3 (3 + 1) = 4
41elexi 3199 . . . 4 3 ∈ V
52elexi 3199 . . . 4 4 ∈ V
6 eleq1 2686 . . . . 5 (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ))
7 oveq1 6611 . . . . . 6 (𝑥 = 3 → (𝑥 + 1) = (3 + 1))
87eqeq1d 2623 . . . . 5 (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦))
96, 83anbi13d 1398 . . . 4 (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦)))
10 eleq1 2686 . . . . 5 (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ))
11 eqeq2 2632 . . . . 5 (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4))
1210, 113anbi23d 1399 . . . 4 (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)))
13 eqid 2621 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}
144, 5, 9, 12, 13brab 4958 . . 3 (3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))
151, 2, 3, 14mpbir3an 1242 . 2 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4
16 breq 4615 . 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 248 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613  {copab 4672  (class class class)co 6604  cc 9878  1c1 9881   + caddc 9883  3c3 11015  4c4 11016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-i2m1 9948  ax-1ne0 9949  ax-rrecex 9952  ax-cnre 9953
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-iota 5810  df-fv 5855  df-ov 6607  df-2 11023  df-3 11024  df-4 11025
This theorem is referenced by: (None)
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