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| Description: Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| dfec2 | ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ec 8747 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
| 2 | imasng 6102 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 {csn 4626 class class class wbr 5143 “ cima 5688 [cec 8743 | 
| 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 | 
| 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-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 | 
| This theorem is referenced by: elqsecl 8811 eqglact 19197 tgpconncompeqg 24120 fvline 36145 ellines 36153 ecres2 38280 | 
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