Theorem eleccnvep 39410

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 6111	. . 3 ⊢ Rel ◡ E
2		relelec 8790	. . 3 ⊢ (Rel ◡ E → (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐵 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝐵)
4		brcnvep 39393	. 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 2147   class class class wbr 5106   E cep 5554  ^◡ccnv 5654  Rel wrel 5660  [cec 8740

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744

This theorem is used by:  eccnvep  39411