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Mirrors > Home > ILE Home > Th. List > ecexg | GIF version |
Description: An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.) |
Ref | Expression |
---|---|
ecexg | ⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ec 6399 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imaexg 4863 | . 2 ⊢ (𝑅 ∈ 𝐵 → (𝑅 “ {𝐴}) ∈ V) | |
3 | 1, 2 | eqeltrid 2204 | 1 ⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 Vcvv 2660 {csn 3497 “ cima 4512 [cec 6395 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-ec 6399 |
This theorem is referenced by: ecelqsg 6450 uniqs 6455 eroveu 6488 th3q 6502 dmaddpq 7155 dmmulpq 7156 addnnnq0 7225 mulnnnq0 7226 addsrpr 7521 mulsrpr 7522 |
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