Theorem eccnvep 38792

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 38791	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]◡ E ↔ 𝑥 ∈ 𝐴))
2	1	eqrdv 2762	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   E cep 5548  ^◡ccnv 5648  [cec 8678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-eprel 5549  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682

This theorem is referenced by:  extep  38793  disjeccnvep  38794  eccnvepres2  38795  ecxrncnvep2  38914  dfeldisj5  39317