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Theorem eccnvep 37661
Description: The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
Assertion
Ref Expression
eccnvep (𝐴𝑉 → [𝐴] E = 𝐴)

Proof of Theorem eccnvep
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleccnvep 37660 . 2 (𝐴𝑉 → (𝑥 ∈ [𝐴] E ↔ 𝑥𝐴))
21eqrdv 2724 1 (𝐴𝑉 → [𝐴] E = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098   E cep 5572  ccnv 5668  [cec 8700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704
This theorem is referenced by:  extep  37662  disjeccnvep  37663  eccnvepres2  37664  dfeldisj5  38102
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