Theorem eleccnvep 38490

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 6064	. . 3 ⊢ Rel ◡ E
2		relelec 8685	. . 3 ⊢ (Rel ◡ E → (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵)
4		brcnvep 38473	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴))
5	3, 4	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡ E ↔ 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114   class class class wbr 5099   E cep 5524  ^◡ccnv 5624  Rel wrel 5630  [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:  eccnvep  38491