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| Mirrors > Home > MPE Home > Th. List > ecelqsi | Structured version Visualization version GIF version | ||
| Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| ecelqsi.1 | ⊢ 𝑅 ∈ V | 
| Ref | Expression | 
|---|---|
| ecelqsi | ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ecelqsi.1 | . 2 ⊢ 𝑅 ∈ V | |
| 2 | ecelqsg 8812 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 [cec 8743 / cqs 8744 | 
| 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 df-qs 8751 | 
| This theorem is referenced by: ecopqsi 8814 addsrpr 11115 mulsrpr 11116 0r 11120 1sr 11121 m1r 11122 addclsr 11123 mulclsr 11124 quseccl0 19203 ghmqusnsglem1 19298 ghmquskerlem1 19301 ghmquskerco 19302 ghmqusker 19305 orbsta 19331 frgpeccl 19779 rngqiprngimf 21307 qustgphaus 24131 vitalilem2 25644 vitalilem3 25645 rloccring 33274 rloc0g 33275 rloc1r 33276 rlocf1 33277 fracfld 33310 nsgqusf1olem1 33441 qsidomlem1 33480 qsdrngilem 33522 qsdrngi 33523 qsdrnglem2 33524 zringfrac 33582 pstmfval 33895 | 
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