Theorem iunid 4022

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 3742	. . . . 5 ⊢ {x} = {y ∣ y = x}
2		equcom 1680	. . . . . 6 ⊢ (y = x ↔ x = y)
3	2	abbii 2466	. . . . 5 ⊢ {y ∣ y = x} = {y ∣ x = y}
4	1, 3	eqtri 2373	. . . 4 ⊢ {x} = {y ∣ x = y}
5	4	a1i 10	. . 3 ⊢ (x ∈ A → {x} = {y ∣ x = y})
6	5	iuneq2i 3988	. 2 ⊢ ∪x ∈ A {x} = ∪x ∈ A {y ∣ x = y}
7		iunab 4013	. . 3 ⊢ ∪x ∈ A {y ∣ x = y} = {y ∣ ∃x ∈ A x = y}
8		risset 2662	. . . 4 ⊢ (y ∈ A ↔ ∃x ∈ A x = y)
9	8	abbii 2466	. . 3 ⊢ {y ∣ y ∈ A} = {y ∣ ∃x ∈ A x = y}
10		abid2 2471	. . 3 ⊢ {y ∣ y ∈ A} = A
11	7, 9, 10	3eqtr2i 2379	. 2 ⊢ ∪x ∈ A {y ∣ x = y} = A
12	6, 11	eqtri 2373	1 ⊢ ∪x ∈ A {x} = A

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  {csn 3738  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-sn 3742  df-iun 3972

This theorem is referenced by:  iunxpconst  4820  uniqs  5985