Theorem iunid 4986

Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Assertion

Ref	Expression
iunid	⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem iunid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sn 4559	. . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
2		equcom 2022	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	abbii 2809	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
4	1, 3	eqtri 2766	. . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
5	4	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦})
6	5	iuneq2i 4942	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦}
7		iunab 4977	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
8		risset 3193	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
9	8	abbii 2809	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
10		abid2 2881	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
11	7, 9, 10	3eqtr2i 2772	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴
12	6, 11	eqtri 2766	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {csn 4558  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-v 3424  df-in 3890  df-ss 3900  df-sn 4559  df-iun 4923

This theorem is referenced by:  iunxpconst  5650  fvn0ssdmfun  6934  abnexg  7584  xpexgALT  7797  uniqs  8524  dftrpred4g  9413  rankcf  10464  dprd2da  19560  t1ficld  22386  discmp  22457  xkoinjcn  22746  metnrmlem2  23929  ovoliunlem1  24571  i1fima  24747  i1fd  24750  itg1addlem5  24770  fnpreimac  30910  gsumpart  31217  elrspunidl  31508  sibfof  32207  bnj1415  32918  cvmlift2lem12  33176  poimirlem30  35734  itg2addnclem2  35756  ftc1anclem6  35782  uniqsALTV  36391  salexct3  43771  salgensscntex  43773  ctvonmbl  44117  vonct  44121