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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfqs2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.) |
| Ref | Expression |
|---|---|
| dfqs2 | ⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-qs 8751 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
| 2 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) | |
| 3 | 2 | rnmpt 5968 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
| 4 | 1, 3 | eqtr4i 2768 | 1 ⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 {cab 2714 ∃wrex 3070 ↦ cmpt 5225 ran crn 5686 [cec 8743 / cqs 8744 |
| 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 df-qs 8751 |
| This theorem is referenced by: qsalrel 42281 |
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