Theorem iunid 5016

Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.)

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-iun 4948	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
2		clel5 3619	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
3		velsn 4596	. . . . 5 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
4	3	rexbii 3083	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
5	2, 4	bitr4i 278	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥})
6	5	eqabi 2871	. 2 ⊢ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
7	1, 6	eqtr4i 2762	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2113  {cab 2714  ∃wrex 3060  {csn 4580  ∪ ciun 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-v 3442  df-sn 4581  df-iun 4948

This theorem is referenced by:  iunxpconst  5697  fvn0ssdmfun  7019  abnexg  7701  xpexgALT  7925  uniqs  8712  rankcf  10690  dprd2da  19975  t1ficld  23273  discmp  23344  xkoinjcn  23633  metnrmlem2  24807  ovoliunlem1  25461  i1fima  25637  i1fd  25640  itg1addlem5  25659  dmdju  32727  fnpreimac  32751  gsumpart  33148  elrspunidl  33511  sibfof  34499  bnj1415  35196  cvmlift2lem12  35510  poimirlem30  37853  itg2addnclem2  37875  ftc1anclem6  37901  salexct3  46607  salgensscntex  46609  ctvonmbl  46954  vonct  46958