Theorem iunab 3694

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 2175	. . . 4 ⊢ ℲyA
2		nfab1 2177	. . . 4 ⊢ Ⅎy{y ∣ φ}
3	1, 2	nfiunxy 3674	. . 3 ⊢ Ⅎy∪ x ∈ A {y ∣ φ}
4		nfab1 2177	. . 3 ⊢ Ⅎy{y ∣ ∃x ∈ A φ}
5	3, 4	cleqf 2198	. 2 ⊢ (∪ x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ} ↔ ∀y(y ∈ ∪ x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ}))
6		abid 2025	. . . 4 ⊢ (y ∈ {y ∣ φ} ↔ φ)
7	6	rexbii 2325	. . 3 ⊢ (∃x ∈ A y ∈ {y ∣ φ} ↔ ∃x ∈ A φ)
8		eliun 3652	. . 3 ⊢ (y ∈ ∪ x ∈ A {y ∣ φ} ↔ ∃x ∈ A y ∈ {y ∣ φ})
9		abid 2025	. . 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 1339	1 ⊢ ∪ x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ}

Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  ∪ ciun 3648

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-iun 3650

This theorem is referenced by:  iunrab  3695  iunid  3703  dfimafn2  5166