Theorem iunab 3988

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 2350	. . . 4 ⊢ Ⅎ𝑦𝐴
2		nfab1 2352	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiunxy 3967	. . 3 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2352	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
5	3, 4	cleqf 2375	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}))
6		abid 2195	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
7	6	rexbii 2515	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
8		eliun 3945	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
9		abid 2195	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
10	7, 8, 9	3bitr4i 212	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
11	5, 10	mpgbir 1477	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ↔ wb 105   = wceq 1373   ∈ wcel 2178  {cab 2193  ∃wrex 2487  ∪ ciun 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-iun 3943

This theorem is referenced by:  iunrab  3989  iunid  3997  dfimafn2  5651