Theorem oprabrexex2 7978

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 7419	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑)}
2		rexcom4 3276	. . . . 5 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3		rexcom4 3276	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
4		rexcom4 3276	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
5		r19.42v 3181	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
6	5	exbii 1842	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
7	4, 6	bitri 274	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
8	7	exbii 1842	. . . . . . 7 ⊢ (∃𝑦∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
9	3, 8	bitri 274	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
10	9	exbii 1842	. . . . 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 275	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑) ↔ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
12	11	abbii 2795	. . 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 2753	. 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 7419	. . . 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 2822	. . 3 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
18	14, 17	abrexex2 7969	. 2 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
19	13, 18	eqeltri 2821	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∃wrex 3060  Vcvv 3463  ⟨cop 4630  {coprab 7416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-v 3465  df-ss 3957  df-uni 4904  df-iun 4993  df-oprab 7419

This theorem is referenced by: (None)