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Mirrors > Home > MPE Home > Th. List > ecexg | Structured version Visualization version 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 8280 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imaexg 7609 | . 2 ⊢ (𝑅 ∈ 𝐵 → (𝑅 “ {𝐴}) ∈ V) | |
3 | 1, 2 | eqeltrid 2914 | 1 ⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 {csn 4557 “ cima 5551 [cec 8276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8280 |
This theorem is referenced by: ecelqsg 8341 uniqs 8346 eroveu 8381 erov 8383 addsrpr 10485 mulsrpr 10486 quslem 16804 eqgen 18271 qusghm 18333 sylow2blem1 18674 vrgpval 18822 znzrhval 20621 qustgpopn 22655 qustgplem 22656 elpi1 23576 pi1xfrval 23585 pi1xfrcnvlem 23587 pi1xfrcnv 23588 pi1cof 23590 pi1coval 23591 tgjustr 26187 qusker 30845 qusvscpbl 30847 qusscaval 30848 pstmfval 31035 fvline 33502 ecex2 35466 qsalrel 39003 |
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