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Theorem dfcoels 34728
 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 34713 . 2 𝐴 = ≀ ( E ↾ 𝐴)
2 1cossres 34727 . 2 ≀ ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)}
3 brcnvep 34578 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
43elv 3418 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
5 brcnvep 34578 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑦𝑦𝑢))
65elv 3418 . . . . 5 (𝑢 E 𝑦𝑦𝑢)
74, 6anbi12i 620 . . . 4 ((𝑢 E 𝑥𝑢 E 𝑦) ↔ (𝑥𝑢𝑦𝑢))
87rexbii 3251 . . 3 (∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦) ↔ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢))
98opabbii 4942 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
101, 2, 93eqtri 2853 1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∃wrex 3118  Vcvv 3414   class class class wbr 4875  {copab 4937   E cep 5256  ◡ccnv 5345   ↾ cres 5348   ≀ ccoss 34519   ∼ ccoels 34520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-eprel 5257  df-xp 5352  df-rel 5353  df-cnv 5354  df-res 5358  df-coss 34712  df-coels 34713 This theorem is referenced by:  brcoels  34733
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