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Theorem dfcoels 37300
Description: Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.)
Assertion
Ref Expression
dfcoels 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Distinct variable group:   𝑢,𝐴,𝑥,𝑦

Proof of Theorem dfcoels
StepHypRef Expression
1 df-coels 37282 . 2 𝐴 = ≀ ( E ↾ 𝐴)
2 1cossres 37299 . 2 ≀ ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)}
3 brcnvep 37133 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
43elv 3481 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
5 brcnvep 37133 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑦𝑦𝑢))
65elv 3481 . . . . 5 (𝑢 E 𝑦𝑦𝑢)
74, 6anbi12i 628 . . . 4 ((𝑢 E 𝑥𝑢 E 𝑦) ↔ (𝑥𝑢𝑦𝑢))
87rexbii 3095 . . 3 (∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦) ↔ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢))
98opabbii 5216 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
101, 2, 93eqtri 2765 1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wrex 3071  Vcvv 3475   class class class wbr 5149  {copab 5211   E cep 5580  ccnv 5676  cres 5679  ccoss 37043  ccoels 37044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-res 5689  df-coss 37281  df-coels 37282
This theorem is referenced by:  brcoels  37305
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