Theorem iunid 4947

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 4526	. . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
2		equcom 2025	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	abbii 2863	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
4	1, 3	eqtri 2821	. . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
5	4	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦})
6	5	iuneq2i 4902	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦}
7		iunab 4938	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
8		risset 3226	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
9	8	abbii 2863	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
10		abid2 2932	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
11	7, 9, 10	3eqtr2i 2827	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴
12	6, 11	eqtri 2821	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  {csn 4525  ∪ ciun 4881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-sn 4526  df-iun 4883

This theorem is referenced by:  iunxpconst  5588  fvn0ssdmfun  6819  abnexg  7458  xpexgALT  7664  uniqs  8340  rankcf  10188  dprd2da  19157  t1ficld  21932  discmp  22003  xkoinjcn  22292  metnrmlem2  23465  ovoliunlem1  24106  i1fima  24282  i1fd  24285  itg1addlem5  24304  fnpreimac  30434  gsumpart  30740  elrspunidl  31014  sibfof  31708  bnj1415  32420  cvmlift2lem12  32674  dftrpred4g  33186  poimirlem30  35087  itg2addnclem2  35109  ftc1anclem6  35135  uniqsALTV  35746  salexct3  42982  salgensscntex  42984  ctvonmbl  43328  vonct  43332