Theorem oprabrexex2 8004

Description: Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Hypotheses

Ref	Expression
oprabrexex2.1	⊢ 𝐴 ∈ V
oprabrexex2.2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V

Assertion

Ref	Expression
oprabrexex2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V

Distinct variable group:   𝑥,𝐴,𝑦,𝑧,𝑤

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

Proof of Theorem oprabrexex2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 7436	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑)}
2		rexcom4 3287	. . . . 5 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3		rexcom4 3287	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
4		rexcom4 3287	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
5		r19.42v 3190	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
6	5	exbii 1847	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
7	4, 6	bitri 275	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
8	7	exbii 1847	. . . . . . 7 ⊢ (∃𝑦∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
9	3, 8	bitri 275	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
10	9	exbii 1847	. . . . 5 ⊢ (∃𝑥∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
11	2, 10	bitr2i 276	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑) ↔ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
12	11	abbii 2808	. . 3 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑)} = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
13	1, 12	eqtri 2764	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
14		oprabrexex2.1	. . 3 ⊢ 𝐴 ∈ V
15		df-oprab 7436	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
16		oprabrexex2.2	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V
17	15, 16	eqeltrri 2837	. . 3 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
18	14, 17	abrexex2 7995	. 2 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
19	13, 18	eqeltri 2836	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713  ∃wrex 3069  Vcvv 3479  ⟨cop 4631  {coprab 7433

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-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-ss 3967  df-uni 4907  df-iun 4992  df-oprab 7436

This theorem is referenced by: (None)