Theorem oprabrexex2 7794

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 7259	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑)}
2		rexcom4 3179	. . . . 5 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3		rexcom4 3179	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
4		rexcom4 3179	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
5		r19.42v 3276	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ 𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤 ∈ 𝐴 𝜑))
6	5	exbii 1851	. . . . . . . . 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 1851	. . . . . . 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 1851	. . . . 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 2809	. . 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 2766	. 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 7259	. . . 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 2836	. . 3 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
18	14, 17	abrexex2 7785	. 2 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
19	13, 18	eqeltri 2835	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422  ⟨cop 4564  {coprab 7256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-oprab 7259

This theorem is referenced by: (None)