Theorem eccnvep 39221

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 39220	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]◡ E ↔ 𝑥 ∈ 𝐴))
2	1	eqrdv 2758	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   E cep 5549  ^◡ccnv 5649  [cec 8715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8719

This theorem is used by:  extep  39222  disjeccnvep  39223  eccnvepres2  39224  ecxrncnvep2  39343  dfeldisj5  39746