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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 8016 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imaexg 7370 | . 2 ⊢ (𝑅 ∈ 𝐵 → (𝑅 “ {𝐴}) ∈ V) | |
3 | 1, 2 | syl5eqel 2910 | 1 ⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 Vcvv 3414 {csn 4399 “ cima 5349 [cec 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-xp 5352 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-ec 8016 |
This theorem is referenced by: ecelqsg 8072 uniqs 8077 eroveu 8113 erov 8115 addsrpr 10219 mulsrpr 10220 quslem 16563 eqgen 18005 qusghm 18055 sylow2blem1 18393 vrgpval 18540 znzrhval 20261 qustgpopn 22300 qustgplem 22301 elpi1 23221 pi1xfrval 23230 pi1xfrcnvlem 23232 pi1xfrcnv 23233 pi1cof 23235 pi1coval 23236 tgjustr 25793 pstmfval 30480 fvline 32785 ecex2 34643 |
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