Theorem eccnvep 38266

Description: The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)

Assertion

Ref	Expression
eccnvep	⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Proof of Theorem eccnvep

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleccnvep 38265	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]◡ E ↔ 𝑥 ∈ 𝐴))
2	1	eqrdv 2727	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   E cep 5518  ^◡ccnv 5618  [cec 8623

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8627

This theorem is referenced by:  extep  38267  disjeccnvep  38268  eccnvepres2  38269  dfeldisj5  38709