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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 8810 | . 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 2105 Vcvv 3477 [cec 8741 / cqs 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 df-qs 8749 |
This theorem is referenced by: ecopqsi 8812 addsrpr 11112 mulsrpr 11113 0r 11117 1sr 11118 m1r 11119 addclsr 11120 mulclsr 11121 quseccl0 19215 ghmqusnsglem1 19310 ghmquskerlem1 19313 ghmquskerco 19314 ghmqusker 19317 orbsta 19343 frgpeccl 19793 rngqiprngimf 21324 qustgphaus 24146 vitalilem2 25657 vitalilem3 25658 rloccring 33256 rloc0g 33257 rloc1r 33258 rlocf1 33259 fracfld 33289 nsgqusf1olem1 33420 qsidomlem1 33459 qsdrngilem 33501 qsdrngi 33502 qsdrnglem2 33503 zringfrac 33561 pstmfval 33856 |
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