Theorem eccnvep 38491

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 38490	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]◡ E ↔ 𝑥 ∈ 𝐴))
2	1	eqrdv 2735	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   E cep 5524  ^◡ccnv 5624  [cec 8635

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639

This theorem is referenced by:  extep  38492  disjeccnvep  38493  eccnvepres2  38494  ecxrncnvep2  38613  dfeldisj5  39016