Theorem iunid 5019

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 4953	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
2		clel5 3619	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
3		velsn 4600	. . . . 5 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
4	3	rexbii 3110	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝑥)
5	2, 4	bitr4i 281	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥})
6	5	eqabi 2896	. 2 ⊢ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑥}}
7	1, 6	eqtr4i 2787	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087  {csn 4584  ∪ ciun 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088  df-v 3453  df-sn 4585  df-iun 4953

This theorem is used by:  iunxpconst  5724  fvn0ssdmfun  7074  abnexg  7770  xpexgALT  7993  uniqs  8794  rankcf  10862  dprd2da  20258  t1ficld  23645  discmp  23716  xkoinjcn  24006  metnrmlem2  25180  ovoliunlem1  25823  i1fima  25999  i1fd  26002  itg1addlem5  26021  rnplynfin  26630  dmdju  33241  fnpreimac  33264  gsumpart  33624  elrspunidl  33978  sibfof  34972  bnj1415  35668  1enumen  35723  cvmlift2lem12  36079  poimirlem30  38568  itg2addnclem2  38590  ftc1anclem6  38616  salexct3  47351  salgensscntex  47353  ctvonmbl  47698  vonct  47702