Theorem iunid 5083

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 5017	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
2		clel5 3678	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
3		velsn 4664	. . . . 5 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
4	3	rexbii 3100	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
5	2, 4	bitr4i 278	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥})
6	5	eqabi 2880	. 2 ⊢ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
7	1, 6	eqtr4i 2771	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  {csn 4648  ∪ ciun 5015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-sn 4649  df-iun 5017

This theorem is referenced by:  iunxpconst  5772  fvn0ssdmfun  7108  abnexg  7791  xpexgALT  8022  uniqs  8835  rankcf  10846  dprd2da  20086  t1ficld  23356  discmp  23427  xkoinjcn  23716  metnrmlem2  24901  ovoliunlem1  25556  i1fima  25732  i1fd  25735  itg1addlem5  25755  fnpreimac  32689  gsumpart  33038  elrspunidl  33421  sibfof  34305  bnj1415  35014  cvmlift2lem12  35282  poimirlem30  37610  itg2addnclem2  37632  ftc1anclem6  37658  uniqsALTV  38285  salexct3  46263  salgensscntex  46265  ctvonmbl  46610  vonct  46614