Theorem iunab 5013

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 2926	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfab1 2928	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiun 4985	. 2 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2928	. 2 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
5		abid 2745	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
6	5	rexbii 3113	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
7		eliun 4958	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
8		abid 2745	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
9	6, 7, 8	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})
10	3, 4, 9	eqri 3933	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146  {cab 2741  ∃wrex 3090  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-iun 4956

This theorem is used by:  iunrab  5014  dfimafn2  6959  pzriprnglem10  21811  pzriprnglem11  21812  rabiun  38664  dfaimafn2  48378  rnfdmpr  48493