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Mirrors > Home > MPE Home > Th. List > iunid | Structured version Visualization version GIF version |
Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
iunid | ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sn 4558 | . . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥} | |
2 | equcom 2016 | . . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | abbii 2883 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
4 | 1, 3 | eqtri 2841 | . . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}) |
6 | 5 | iuneq2i 4931 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} |
7 | iunab 4966 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} | |
8 | risset 3264 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
9 | 8 | abbii 2883 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} |
10 | abid2 2954 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
11 | 7, 9, 10 | 3eqtr2i 2847 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴 |
12 | 6, 11 | eqtri 2841 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 {cab 2796 ∃wrex 3136 {csn 4557 ∪ ciun 4910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-in 3940 df-ss 3949 df-sn 4558 df-iun 4912 |
This theorem is referenced by: iunxpconst 5617 fvn0ssdmfun 6834 abnexg 7467 xpexgALT 7671 uniqs 8346 rankcf 10187 dprd2da 19093 t1ficld 21863 discmp 21934 xkoinjcn 22223 metnrmlem2 23395 ovoliunlem1 24030 i1fima 24206 i1fd 24209 itg1addlem5 24228 fnpreimac 30344 sibfof 31497 bnj1415 32207 cvmlift2lem12 32458 dftrpred4g 32970 poimirlem30 34803 itg2addnclem2 34825 ftc1anclem6 34853 uniqsALTV 35467 salexct3 42502 salgensscntex 42504 ctvonmbl 42848 vonct 42852 |
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