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Mirrors > Home > ILE Home > Th. List > ecqs | GIF version |
Description: Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.) |
Ref | Expression |
---|---|
ecqs.1 | ⊢ 𝑅 ∈ V |
Ref | Expression |
---|---|
ecqs | ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ec 6511 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | ecqs.1 | . . 3 ⊢ 𝑅 ∈ V | |
3 | uniqs 6567 | . . 3 ⊢ (𝑅 ∈ V → ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴})) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴}) |
5 | 1, 4 | eqtr4i 2194 | 1 ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 Vcvv 2730 {csn 3581 ∪ cuni 3794 “ cima 4612 [cec 6507 / cqs 6508 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-ec 6511 df-qs 6515 |
This theorem is referenced by: (None) |
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