Theorem iunid 5024

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 4957	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
2		clel5 3631	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
3		velsn 4605	. . . . 5 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
4	3	rexbii 3076	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
5	2, 4	bitr4i 278	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥})
6	5	eqabi 2863	. 2 ⊢ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
7	1, 6	eqtr4i 2755	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  {csn 4589  ∪ ciun 4955

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-v 3449  df-sn 4590  df-iun 4957

This theorem is referenced by:  iunxpconst  5711  fvn0ssdmfun  7046  abnexg  7732  xpexgALT  7960  uniqs  8747  rankcf  10730  dprd2da  19974  t1ficld  23214  discmp  23285  xkoinjcn  23574  metnrmlem2  24749  ovoliunlem1  25403  i1fima  25579  i1fd  25582  itg1addlem5  25601  dmdju  32571  fnpreimac  32595  gsumpart  32997  elrspunidl  33399  sibfof  34331  bnj1415  35028  cvmlift2lem12  35301  poimirlem30  37644  itg2addnclem2  37666  ftc1anclem6  37692  salexct3  46340  salgensscntex  46342  ctvonmbl  46687  vonct  46691