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Theorem dfcoels 35783
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 35768 . 2 𝐴 = ≀ ( E ↾ 𝐴)
2 1cossres 35782 . 2 ≀ ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)}
3 brcnvep 35634 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
43elv 3485 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
5 brcnvep 35634 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑦𝑦𝑢))
65elv 3485 . . . . 5 (𝑢 E 𝑦𝑦𝑢)
74, 6anbi12i 629 . . . 4 ((𝑢 E 𝑥𝑢 E 𝑦) ↔ (𝑥𝑢𝑦𝑢))
87rexbii 3241 . . 3 (∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦) ↔ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢))
98opabbii 5119 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
101, 2, 93eqtri 2851 1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wrex 3134  Vcvv 3480   class class class wbr 5052  {copab 5114   E cep 5451  ccnv 5541  cres 5544  ccoss 35561  ccoels 35562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-eprel 5452  df-xp 5548  df-rel 5549  df-cnv 5550  df-res 5554  df-coss 35767  df-coels 35768
This theorem is referenced by:  brcoels  35788
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