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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfqs3 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.) |
| Ref | Expression |
|---|---|
| dfqs3 | ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-qs 8751 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
| 2 | iunsn 5066 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
| 3 | 1, 2 | eqtr4i 2768 | 1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 {cab 2714 ∃wrex 3070 {csn 4626 ∪ ciun 4991 [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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-sn 4627 df-iun 4993 df-qs 8751 |
| This theorem is referenced by: prjspval2 42623 |
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