![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ecelqsdm | GIF version |
Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.) |
Ref | Expression |
---|---|
ecelqsdm | ⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsn0m 6400 | . . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ [𝐵]𝑅) | |
2 | ecdmn0m 6374 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅) | |
3 | 1, 2 | sylibr 133 | . 2 ⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ dom 𝑅) |
4 | simpl 108 | . 2 ⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → dom 𝑅 = 𝐴) | |
5 | 3, 4 | eleqtrd 2173 | 1 ⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∃wex 1433 ∈ wcel 1445 dom cdm 4467 [cec 6330 / cqs 6331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-xp 4473 df-cnv 4475 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-ec 6334 df-qs 6338 |
This theorem is referenced by: th3qlem1 6434 nnnq0lem1 7102 prsrlem1 7385 gt0srpr 7391 |
Copyright terms: Public domain | W3C validator |