Theorem ecid 8345

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 3444	. . . 4 ⊢ 𝑦 ∈ V
2		ecid.1	. . . 4 ⊢ 𝐴 ∈ V
3	1, 2	elec 8316	. . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)
4	2, 1	brcnv 5717	. . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)
5	2	epeli 5432	. . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
6	3, 4, 5	3bitri 300	. 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴)
7	6	eqriv 2795	1 ⊢ [𝐴]◡ E = 𝐴

Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030   E cep 5429  ^◡ccnv 5518  [cec 8270

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ec 8274

This theorem is referenced by:  qsid  8346  addcnsrec  10554  mulcnsrec  10555