Theorem eleccnvep 39022

Description: Elementhood in the converse epsilon coset of 𝐴 is elementhood in 𝐴. (Contributed by Peter Mazsa, 27-Jan-2019.)

Assertion

Ref	Expression
eleccnvep	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡ E ↔ 𝐵 ∈ 𝐴))

Proof of Theorem eleccnvep

Step	Hyp	Ref	Expression
1		relcnv 6104	. . 3 ⊢ Rel ◡ E
2		relelec 8747	. . 3 ⊢ (Rel ◡ E → (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵)
4		brcnvep 39005	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴))
5	3, 4	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡ E ↔ 𝐵 ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145   class class class wbr 5107   E cep 5558  ^◡ccnv 5658  Rel wrel 5664  [cec 8697

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8701

This theorem is used by:  eccnvep  39023