Theorem iunab 5055

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 2902	. . . 4 ⊢ Ⅎ𝑦𝐴
2		nfab1 2904	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiun 5027	. . 3 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2904	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
5	3, 4	cleqf 2931	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}))
6		abid 2715	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
7	6	rexbii 3091	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
8		eliun 4999	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
9		abid 2715	. . 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 1795	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ↔ wb 206   = wceq 1536   ∈ wcel 2105  {cab 2711  ∃wrex 3067  ∪ ciun 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-v 3479  df-iun 4997

This theorem is referenced by:  iunrab  5056  iunidOLD  5065  dfimafn2  6971  pzriprnglem10  21518  pzriprnglem11  21519  rabiun  37579  dfaimafn2  47115  rnfdmpr  47230