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Theorem ex-opab 27521
Description: Example for df-opab 4821. 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 11208 . . 3 3 ∈ ℂ
2 4cn 11211 . . 3 4 ∈ ℂ
3 3p1e4 11266 . . 3 (3 + 1) = 4
41elexi 3317 . . . 4 3 ∈ V
52elexi 3317 . . . 4 4 ∈ V
6 eleq1 2791 . . . . 5 (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ))
7 oveq1 6772 . . . . . 6 (𝑥 = 3 → (𝑥 + 1) = (3 + 1))
87eqeq1d 2726 . . . . 5 (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦))
96, 83anbi13d 1514 . . . 4 (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦)))
10 eleq1 2791 . . . . 5 (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ))
11 eqeq2 2735 . . . . 5 (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4))
1210, 113anbi23d 1515 . . . 4 (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)))
13 eqid 2724 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}
144, 5, 9, 12, 13brab 5102 . . 3 (3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))
151, 2, 3, 14mpbir3an 1381 . 2 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4
16 breq 4762 . 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 1072   = wceq 1596  wcel 2103   class class class wbr 4760  {copab 4820  (class class class)co 6765  cc 10047  1c1 10050   + caddc 10052  3c3 11184  4c4 11185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-i2m1 10117  ax-1ne0 10118  ax-rrecex 10121  ax-cnre 10122
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-iota 5964  df-fv 6009  df-ov 6768  df-2 11192  df-3 11193  df-4 11194
This theorem is referenced by: (None)
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