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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 8278 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
2 | iunsn 39412 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
3 | 1, 2 | eqtr4i 2824 | 1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 {cab 2776 ∃wrex 3107 {csn 4525 ∪ ciun 4881 [cec 8270 / cqs 8271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rex 3112 df-v 3443 df-sn 4526 df-iun 4883 df-qs 8278 |
This theorem is referenced by: prjspval2 39607 |
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