Theorem iunab 4981

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 2907	. . . 4 ⊢ Ⅎ𝑦𝐴
2		nfab1 2909	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiun 4954	. . 3 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2909	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
5	3, 4	cleqf 2938	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}))
6		abid 2719	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
7	6	rexbii 3181	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
8		eliun 4928	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
9		abid 2719	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
11	5, 10	mpgbir 1802	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-iun 4926

This theorem is referenced by:  iunrab  4982  iunid  4990  dfimafn2  6833  rabiun  35750  dfaimafn2  44658  rnfdmpr  44773