Theorem iunab 4013

Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)

Assertion

Ref	Expression
iunab	⊢ ∪x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ}

Distinct variable groups:   y,A   x,y

Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem iunab

Step	Hyp	Ref	Expression
1		nfcv 2490	. . . 4 ⊢ ℲyA
2		nfab1 2492	. . . 4 ⊢ Ⅎy{y ∣ φ}
3	1, 2	nfiun 3996	. . 3 ⊢ Ⅎy∪x ∈ A {y ∣ φ}
4		nfab1 2492	. . 3 ⊢ Ⅎy{y ∣ ∃x ∈ A φ}
5	3, 4	cleqf 2514	. 2 ⊢ (∪x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ} ↔ ∀y(y ∈ ∪x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ}))
6		abid 2341	. . . 4 ⊢ (y ∈ {y ∣ φ} ↔ φ)
7	6	rexbii 2640	. . 3 ⊢ (∃x ∈ A y ∈ {y ∣ φ} ↔ ∃x ∈ A φ)
8		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A {y ∣ φ} ↔ ∃x ∈ A y ∈ {y ∣ φ})
9		abid 2341	. . 3 ⊢ (y ∈ {y ∣ ∃x ∈ A φ} ↔ ∃x ∈ A φ)
10	7, 8, 9	3bitr4i 268	. 2 ⊢ (y ∈ ∪x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ})
11	5, 10	mpgbir 1550	1 ⊢ ∪x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ}

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ∪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-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-iun 3972

This theorem is referenced by:  iunrab  4014  iunid  4022  dfimafn2  5368