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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 2554 | . . . 4 ⊢ y ∈ V | |
2 | elecg 6080 | . . . 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 4459 | . . . 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 4018 | . . 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 2035 | 1 ⊢ (A ∈ 𝑉 → [A]◡ E = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 Vcvv 2551 class class class wbr 3755 E cep 4015 ◡ccnv 4287 [cec 6040 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-eprel 4017 df-xp 4294 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-ec 6044 |
This theorem is referenced by: addcnsrec 6739 mulcnsrec 6740 |
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