Theorem iunid 3863

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 3528	. . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
2		equcom 1682	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	abbii 2253	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
4	1, 3	eqtri 2158	. . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
5	4	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦})
6	5	iuneq2i 3826	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦}
7		iunab 3854	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
8		risset 2461	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
9	8	abbii 2253	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
10		abid2 2258	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
11	7, 9, 10	3eqtr2i 2164	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴
12	6, 11	eqtri 2158	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1331   ∈ wcel 1480  {cab 2123  ∃wrex 2415  {csn 3522  ∪ ciun 3808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-sn 3528  df-iun 3810

This theorem is referenced by:  abnexg  4362  iunxpconst  4594  xpexgALT  6024  uniqs  6480