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Mirrors > Home > ILE Home > Th. List > ecidg | GIF version |
Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.) |
Ref | Expression |
---|---|
ecidg | ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | elecg 6144 | . . . 4 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)) | |
3 | 1, 2 | mpan 400 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)) |
4 | brcnvg 4516 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)) | |
5 | 1, 4 | mpan2 401 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)) |
6 | epelg 4027 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
7 | 3, 5, 6 | 3bitrd 203 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴)) |
8 | 7 | eqrdv 2038 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 class class class wbr 3764 E cep 4024 ◡ccnv 4344 [cec 6104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-eprel 4026 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-ec 6108 |
This theorem is referenced by: addcnsrec 6918 mulcnsrec 6919 |
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