Theorem iunab 3677

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 2160	. . . 4 ⊢ ℲyA
2		nfab1 2162	. . . 4 ⊢ Ⅎy{y ∣ φ}
3	1, 2	nfiunxy 3657	. . 3 ⊢ Ⅎy∪ x ∈ A {y ∣ φ}
4		nfab1 2162	. . 3 ⊢ Ⅎy{y ∣ ∃x ∈ A φ}
5	3, 4	cleqf 2183	. 2 ⊢ (∪ x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ} ↔ ∀y(y ∈ ∪ x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ}))
6		abid 2010	. . . 4 ⊢ (y ∈ {y ∣ φ} ↔ φ)
7	6	rexbii 2309	. . 3 ⊢ (∃x ∈ A y ∈ {y ∣ φ} ↔ ∃x ∈ A φ)
8		eliun 3635	. . 3 ⊢ (y ∈ ∪ x ∈ A {y ∣ φ} ↔ ∃x ∈ A y ∈ {y ∣ φ})
9		abid 2010	. . 3 ⊢ (y ∈ {y ∣ ∃x ∈ A φ} ↔ ∃x ∈ A φ)
10	7, 8, 9	3bitr4i 201	. 2 ⊢ (y ∈ ∪ x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ})
11	5, 10	mpgbir 1322	1 ⊢ ∪ x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ}

Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  ∃wrex 2285  ∪ ciun 3631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-iun 3633

This theorem is referenced by:  iunrab  3678  iunid  3686  dfimafn2  5148