Theorem iunid 3928

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 3589	. . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
2		equcom 1699	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	abbii 2286	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
4	1, 3	eqtri 2191	. . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}
5	4	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦})
6	5	iuneq2i 3891	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦}
7		iunab 3919	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
8		risset 2498	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
9	8	abbii 2286	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦}
10		abid2 2291	. . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
11	7, 9, 10	3eqtr2i 2197	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴
12	6, 11	eqtri 2191	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

Syntax hints:   = wceq 1348   ∈ wcel 2141  {cab 2156  ∃wrex 2449  {csn 3583  ∪ ciun 3873

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-iun 3875

This theorem is referenced by:  abnexg  4431  iunxpconst  4671  xpexgALT  6112  uniqs  6571