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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 4398 | . . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥} | |
2 | equcom 2122 | . . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | abbii 2944 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
4 | 1, 3 | eqtri 2849 | . . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}) |
6 | 5 | iuneq2i 4759 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} |
7 | iunab 4786 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} | |
8 | risset 3272 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
9 | 8 | abbii 2944 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} |
10 | abid2 2950 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
11 | 7, 9, 10 | 3eqtr2i 2855 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴 |
12 | 6, 11 | eqtri 2849 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 {cab 2811 ∃wrex 3118 {csn 4397 ∪ ciun 4740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-v 3416 df-in 3805 df-ss 3812 df-sn 4398 df-iun 4742 |
This theorem is referenced by: iunxpconst 5408 fvn0ssdmfun 6599 abnexg 7225 xpexgALT 7421 uniqs 8072 rankcf 9914 dprd2da 18795 t1ficld 21502 discmp 21572 xkoinjcn 21861 metnrmlem2 23033 ovoliunlem1 23668 i1fima 23844 i1fd 23847 itg1addlem5 23866 sibfof 30936 bnj1415 31641 cvmlift2lem12 31831 dftrpred4g 32261 poimirlem30 33976 itg2addnclem2 33998 ftc1anclem6 34026 uniqsALTV 34642 salexct3 41344 salgensscntex 41346 ctvonmbl 41690 vonct 41694 |
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