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Mirrors > Home > MPE Home > Th. List > ecqs | Structured version Visualization version GIF version |
Description: Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.) |
Ref | Expression |
---|---|
ecqs.1 | ⊢ 𝑅 ∈ V |
Ref | Expression |
---|---|
ecqs | ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ec 8293 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | ecqs.1 | . . 3 ⊢ 𝑅 ∈ V | |
3 | uniqs 8359 | . . 3 ⊢ (𝑅 ∈ V → ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴})) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴}) |
5 | 1, 4 | eqtr4i 2849 | 1 ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 ∪ cuni 4840 “ cima 5560 [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-iun 4923 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: (None) |
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