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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 2604 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | elecg 6167 | . . . 4 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)) | |
3 | 1, 2 | mpan 414 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)) |
4 | brcnvg 4534 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)) | |
5 | 1, 4 | mpan2 415 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)) |
6 | epelg 4045 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
7 | 3, 5, 6 | 3bitrd 212 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴)) |
8 | 7 | eqrdv 2079 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 Vcvv 2601 class class class wbr 3785 E cep 4042 ◡ccnv 4362 [cec 6127 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-eprel 4044 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-ec 6131 |
This theorem is referenced by: addcnsrec 7010 mulcnsrec 7011 |
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