Theorem iunopabOLD 5547

Description: Obsolete version of iunopab 5546 as of 11-Dec-2024. (Contributed by Stefan O'Rear, 20-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
iunopabOLD	⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝑧   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑧)

Proof of Theorem iunopabOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elopab 5514	. . . . 5 ⊢ (𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2	1	rexbii 3082	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		rexcom4 3273	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		rexcom4 3273	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		r19.42v 3178	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
6	5	exbii 1847	. . . . . . 7 ⊢ (∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
7	4, 6	bitri 275	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
8	7	exbii 1847	. . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
9	3, 8	bitri 275	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
10	2, 9	bitri 275	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
11	10	abbii 2801	. 2 ⊢ {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑)}
12		df-iun 4975	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}}
13		df-opab 5188	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑)}
14	11, 12, 13	3eqtr4i 2767	1 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  ∃wrex 3059  ⟨cop 4614  ∪ ciun 4973  {copab 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rex 3060  df-v 3466  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-iun 4975  df-opab 5188

This theorem is referenced by: (None)