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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 3497 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | ecid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | elec 8327 | . . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦) |
4 | 2, 1 | brcnv 5747 | . . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴) |
5 | 2 | epeli 5462 | . . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
6 | 3, 4, 5 | 3bitri 299 | . 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴) |
7 | 6 | eqriv 2818 | 1 ⊢ [𝐴]◡ E = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 E cep 5458 ◡ccnv 5548 [cec 8281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-eprel 5459 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ec 8285 |
This theorem is referenced by: qsid 8357 addcnsrec 10559 mulcnsrec 10560 |
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