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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 8354 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 [cec 8289 / cqs 8290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ec 8293 df-qs 8297 |
This theorem is referenced by: ecopqsi 8356 addsrpr 10499 mulsrpr 10500 0r 10504 1sr 10505 m1r 10506 addclsr 10507 mulclsr 10508 quseccl 18338 orbsta 18445 frgpeccl 18889 qustgphaus 22733 vitalilem2 24212 vitalilem3 24213 qsidomlem1 30967 pstmfval 31138 |
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