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Mirrors > Home > ILE Home > Th. List > ecelqsg | GIF version |
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecelqsg | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . 3 ⊢ [𝐵]𝑅 = [𝐵]𝑅 | |
2 | eceq1 6457 | . . . . 5 ⊢ (𝑥 = 𝐵 → [𝑥]𝑅 = [𝐵]𝑅) | |
3 | 2 | eqeq2d 2149 | . . . 4 ⊢ (𝑥 = 𝐵 → ([𝐵]𝑅 = [𝑥]𝑅 ↔ [𝐵]𝑅 = [𝐵]𝑅)) |
4 | 3 | rspcev 2784 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ [𝐵]𝑅 = [𝐵]𝑅) → ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅) |
5 | 1, 4 | mpan2 421 | . 2 ⊢ (𝐵 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅) |
6 | ecexg 6426 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → [𝐵]𝑅 ∈ V) | |
7 | elqsg 6472 | . . . 4 ⊢ ([𝐵]𝑅 ∈ V → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅)) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅)) |
9 | 8 | biimpar 295 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
10 | 5, 9 | sylan2 284 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 Vcvv 2681 [cec 6420 / cqs 6421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-ec 6424 df-qs 6428 |
This theorem is referenced by: ecelqsi 6476 qliftlem 6500 eroprf 6515 |
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