Theorem ecid 8729

Description: A set is equal to its coset under the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
ecid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ecid	⊢ [𝐴]◡ E = 𝐴

Proof of Theorem ecid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3446	. . . 4 ⊢ 𝑦 ∈ V
2		ecid.1	. . . 4 ⊢ 𝐴 ∈ V
3	1, 2	elec 8692	. . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)
4	2, 1	brcnv 5839	. . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)
5	2	epeli 5534	. . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
6	3, 4, 5	3bitri 297	. 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴)
7	6	eqriv 2734	1 ⊢ [𝐴]◡ E = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100   E cep 5531  ^◡ccnv 5631  [cec 8643

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 5243  ax-pr 5379

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 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5532  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647

This theorem is referenced by:  qsid  8730  addcnsrec  11066  mulcnsrec  11067