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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.) (Proof shortened by SN, 15-Jan-2025.) |
| Ref | Expression |
|---|---|
| iunid | ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-iun 4993 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}} | |
| 2 | clel5 3665 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥) | |
| 3 | velsn 4642 | . . . . 5 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
| 4 | 3 | rexbii 3094 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥) |
| 5 | 2, 4 | bitr4i 278 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}) |
| 6 | 5 | eqabi 2877 | . 2 ⊢ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}} |
| 7 | 1, 6 | eqtr4i 2768 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 {csn 4626 ∪ ciun 4991 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-sn 4627 df-iun 4993 |
| This theorem is referenced by: iunxpconst 5758 fvn0ssdmfun 7094 abnexg 7776 xpexgALT 8006 uniqs 8817 rankcf 10817 dprd2da 20062 t1ficld 23335 discmp 23406 xkoinjcn 23695 metnrmlem2 24882 ovoliunlem1 25537 i1fima 25713 i1fd 25716 itg1addlem5 25735 dmdju 32657 fnpreimac 32681 gsumpart 33060 elrspunidl 33456 sibfof 34342 bnj1415 35052 cvmlift2lem12 35319 poimirlem30 37657 itg2addnclem2 37679 ftc1anclem6 37705 uniqsALTV 38330 salexct3 46357 salgensscntex 46359 ctvonmbl 46704 vonct 46708 |
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