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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 7984 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imaexg 7338 | . 2 ⊢ (𝑅 ∈ 𝐵 → (𝑅 “ {𝐴}) ∈ V) | |
3 | 1, 2 | syl5eqel 2882 | 1 ⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 Vcvv 3385 {csn 4368 “ cima 5315 [cec 7980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ec 7984 |
This theorem is referenced by: ecelqsg 8040 uniqs 8045 eroveu 8081 erov 8083 addsrpr 10184 mulsrpr 10185 quslem 16518 eqgen 17960 qusghm 18010 sylow2blem1 18348 vrgpval 18495 znzrhval 20216 qustgpopn 22251 qustgplem 22252 elpi1 23172 pi1xfrval 23181 pi1xfrcnvlem 23183 pi1xfrcnv 23184 pi1cof 23186 pi1coval 23187 tgjustr 25725 pstmfval 30455 fvline 32764 ecex2 34594 |
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