Theorem eleccnvep 36416

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 6012	. . 3 ⊢ Rel ◡ E
2		relelec 8543	. . 3 ⊢ (Rel ◡ E → (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵)
4		brcnvep 36404	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝐵 ↔ 𝐵 ∈ 𝐴))
5	3, 4	syl5bb 283	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡ E ↔ 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106   class class class wbr 5074   E cep 5494  ^◡ccnv 5588  Rel wrel 5594  [cec 8496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500

This theorem is referenced by:  eccnvep  36417