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Theorem coss2cnvepres 38419
Description: Special case of coss1cnvres 38418. (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 38418 . 2 ( E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑢 E 𝑥𝑣 E 𝑥))}
2 brcnvep 38266 . . . . . . 7 (𝑢 ∈ V → (𝑢 E 𝑥𝑥𝑢))
32elv 3485 . . . . . 6 (𝑢 E 𝑥𝑥𝑢)
4 brcnvep 38266 . . . . . . 7 (𝑣 ∈ V → (𝑣 E 𝑥𝑥𝑣))
54elv 3485 . . . . . 6 (𝑣 E 𝑥𝑥𝑣)
63, 5anbi12i 628 . . . . 5 ((𝑢 E 𝑥𝑣 E 𝑥) ↔ (𝑥𝑢𝑥𝑣))
76exbii 1848 . . . 4 (∃𝑥(𝑢 E 𝑥𝑣 E 𝑥) ↔ ∃𝑥(𝑥𝑢𝑥𝑣))
87anbi2i 623 . . 3 (((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑢 E 𝑥𝑣 E 𝑥)) ↔ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣)))
98opabbii 5210 . 2 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑢 E 𝑥𝑣 E 𝑥))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣))}
101, 9eqtri 2765 1 ( E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ ∃𝑥(𝑥𝑢𝑥𝑣))}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143  {copab 5205   E cep 5583  ccnv 5684  cres 5687  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-res 5697  df-coss 38412
This theorem is referenced by: (None)
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