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Theorem cnvepresex 38316
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 38280 . 2 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
2 id 22 . . 3 (𝐴𝑉𝐴𝑉)
3 abid2 2877 . . . . 5 {𝑦𝑦𝑥} = 𝑥
4 vex 3482 . . . . 5 𝑥 ∈ V
53, 4eqeltri 2835 . . . 4 {𝑦𝑦𝑥} ∈ V
65a1i 11 . . 3 ((𝐴𝑉𝑥𝐴) → {𝑦𝑦𝑥} ∈ V)
72, 6opabex3d 7989 . 2 (𝐴𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} ∈ V)
81, 7eqeltrid 2843 1 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  {cab 2712  Vcvv 3478  {copab 5210   E cep 5588  ccnv 5688  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-res 5701
This theorem is referenced by:  eccnvepex  38317  cnvepimaex  38318  cnvepima  38319  xrncnvepresex  38390  1cosscnvepresex  38403  cnvepresdmqss  38634  eleldisjseldisj  38711  mpets2  38823
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