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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 8746 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | ecqs.1 | . . 3 ⊢ 𝑅 ∈ V | |
3 | uniqs 8816 | . . 3 ⊢ (𝑅 ∈ V → ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴})) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴}) |
5 | 1, 4 | eqtr4i 2766 | 1 ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ∪ cuni 4912 “ cima 5692 [cec 8742 / cqs 8743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 |
This theorem is referenced by: (None) |
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