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Theorem cnvepresex 38335
Description: Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
Assertion
Ref Expression
cnvepresex (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)

Proof of Theorem cnvepresex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvepres 38299 . 2 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
2 id 22 . . 3 (𝐴𝑉𝐴𝑉)
3 abid2 2879 . . . . 5 {𝑦𝑦𝑥} = 𝑥
4 vex 3484 . . . . 5 𝑥 ∈ V
53, 4eqeltri 2837 . . . 4 {𝑦𝑦𝑥} ∈ V
65a1i 11 . . 3 ((𝐴𝑉𝑥𝐴) → {𝑦𝑦𝑥} ∈ V)
72, 6opabex3d 7990 . 2 (𝐴𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} ∈ V)
81, 7eqeltrid 2845 1 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  {cab 2714  Vcvv 3480  {copab 5205   E cep 5583  ccnv 5684  cres 5687
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-nf 1784  df-sb 2065  df-mo 2540  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-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-res 5697
This theorem is referenced by:  eccnvepex  38336  cnvepimaex  38337  cnvepima  38338  xrncnvepresex  38409  1cosscnvepresex  38422  cnvepresdmqss  38653  eleldisjseldisj  38730  mpets2  38842
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