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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 8749 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
2 | iunsn 5070 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
3 | 1, 2 | eqtr4i 2765 | 1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {cab 2711 ∃wrex 3067 {csn 4630 ∪ ciun 4995 [cec 8741 / cqs 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-v 3479 df-sn 4631 df-iun 4997 df-qs 8749 |
This theorem is referenced by: prjspval2 42599 |
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