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Theorem cnvepresex 36031
 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 35995 . 2 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
2 id 22 . . 3 (𝐴𝑉𝐴𝑉)
3 abid2 2894 . . . . 5 {𝑦𝑦𝑥} = 𝑥
4 vex 3413 . . . . 5 𝑥 ∈ V
53, 4eqeltri 2848 . . . 4 {𝑦𝑦𝑥} ∈ V
65a1i 11 . . 3 ((𝐴𝑉𝑥𝐴) → {𝑦𝑦𝑥} ∈ V)
72, 6opabex3d 7670 . 2 (𝐴𝑉 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} ∈ V)
81, 7eqeltrid 2856 1 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  {cab 2735  Vcvv 3409  {copab 5094   E cep 5434  ◡ccnv 5523   ↾ cres 5526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-eprel 5435  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343 This theorem is referenced by:  eccnvepex  36032  cnvepimaex  36033  cnvepima  36034  xrncnvepresex  36096  1cosscnvepresex  36106  cnvepresdmqss  36326  eleldisjseldisj  36402
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