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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 8747 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
| 2 | imaexg 7935 | . 2 ⊢ (𝑅 ∈ 𝐵 → (𝑅 “ {𝐴}) ∈ V) | |
| 3 | 1, 2 | eqeltrid 2845 | 1 ⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 {csn 4626 “ cima 5688 [cec 8743 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 |
| This theorem is referenced by: ecelqsg 8812 uniqs 8817 eroveu 8852 erov 8854 addsrpr 11115 mulsrpr 11116 quslem 17588 eqgen 19199 qusghm 19273 ghmquskerco 19302 sylow2blem1 19638 vrgpval 19785 rngqiprngimf1 21310 znzrhval 21565 qustgpopn 24128 qustgplem 24129 elpi1 25078 pi1xfrval 25087 pi1xfrcnvlem 25089 pi1xfrcnv 25090 pi1cof 25092 pi1coval 25093 tgjustr 28482 rlocf1 33277 qusker 33377 qusvscpbl 33379 qusvsval 33380 qusrn 33437 zringfrac 33582 pstmfval 33895 fvline 36145 ecex2 38329 |
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