Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcoels Structured version   Visualization version   GIF version

Theorem dfcoels 36705
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 36687 . 2 𝐴 = ≀ ( E ↾ 𝐴)
2 1cossres 36704 . 2 ≀ ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)}
3 brcnvep 36538 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
43elv 3447 . . . . 5 (𝑢 E 𝑥𝑥𝑢)
5 brcnvep 36538 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑦𝑦𝑢))
65elv 3447 . . . . 5 (𝑢 E 𝑦𝑦𝑢)
74, 6anbi12i 627 . . . 4 ((𝑢 E 𝑥𝑢 E 𝑦) ↔ (𝑥𝑢𝑦𝑢))
87rexbii 3093 . . 3 (∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦) ↔ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢))
98opabbii 5159 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 E 𝑥𝑢 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
101, 2, 93eqtri 2768 1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1540  wrex 3070  Vcvv 3441   class class class wbr 5092  {copab 5154   E cep 5523  ccnv 5619  cres 5622  ccoss 36446  ccoels 36447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-eprel 5524  df-xp 5626  df-rel 5627  df-cnv 5628  df-res 5632  df-coss 36686  df-coels 36687
This theorem is referenced by:  brcoels  36710
  Copyright terms: Public domain W3C validator