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Theorem coss2cnvepres 38399
Description: Special case of coss1cnvres 38398. (Contributed by Peter Mazsa, 8-Jun-2020.)
Assertion
Ref Expression
coss2cnvepres ( E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣))}
Distinct variable group:   𝑢,𝐴,𝑣,𝑥

Proof of Theorem coss2cnvepres
StepHypRef Expression
1 coss1cnvres 38398 . 2 ( E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑢 E 𝑥𝑣 E 𝑥))}
2 brcnvep 38246 . . . . . . 7 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
32elv 3482 . . . . . 6 (𝑢 E 𝑥𝑥𝑢)
4 brcnvep 38246 . . . . . . 7 (𝑣 ∈ V → (𝑣 E 𝑥𝑥𝑣))
54elv 3482 . . . . . 6 (𝑣 E 𝑥𝑥𝑣)
63, 5anbi12i 628 . . . . 5 ((𝑢 E 𝑥𝑣 E 𝑥) ↔ (𝑥𝑢𝑥𝑣))
76exbii 1844 . . . 4 (∃𝑥(𝑢 E 𝑥𝑣 E 𝑥) ↔ ∃𝑥(𝑥𝑢𝑥𝑣))
87anbi2i 623 . . 3 (((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑢 E 𝑥𝑣 E 𝑥)) ↔ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣)))
98opabbii 5214 . 2 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑢 E 𝑥𝑣 E 𝑥))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣))}
101, 9eqtri 2762 1 ( E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣))}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477   class class class wbr 5147  {copab 5209   E cep 5587  ccnv 5687  cres 5690  ccoss 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-res 5700  df-coss 38392
This theorem is referenced by: (None)
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