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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 8713 | . 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 2761 | 1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 {cab 2707 ∃wrex 3068 {csn 4629 ∪ ciun 4998 [cec 8705 / cqs 8706 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rex 3069 df-v 3474 df-sn 4630 df-iun 5000 df-qs 8713 |
This theorem is referenced by: prjspval2 41659 |
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