![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ecid | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ecid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ecid | ⊢ [𝐴]◡ E = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3474 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | ecid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | elec 8764 | . . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦) |
4 | 2, 1 | brcnv 5880 | . . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴) |
5 | 2 | epeli 5579 | . . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
6 | 3, 4, 5 | 3bitri 297 | . 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴) |
7 | 6 | eqriv 2725 | 1 ⊢ [𝐴]◡ E = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3470 class class class wbr 5143 E cep 5576 ◡ccnv 5672 [cec 8717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-opab 5206 df-eprel 5577 df-xp 5679 df-cnv 5681 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ec 8721 |
This theorem is referenced by: qsid 8796 addcnsrec 11161 mulcnsrec 11162 |
Copyright terms: Public domain | W3C validator |