Theorem dfoprab2 3021

Description: Class abstraction for operations in terms of class abstraction of ordered pairs.

Assertion

Ref	Expression
dfoprab2	⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}

Distinct variable group(s):   x,z,w   y,z,w   φ,w

Proof of Theorem dfoprab2

Step	Hyp	Ref	Expression
1		excom 728	. . . 4 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w∃z∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
2		exrot4 778	. . . . 5 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
3		19.42v 966	. . . . . . 7 ⊢ (∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ ∃w w = ⟨x, y⟩))
4		opeq1 1876	. . . . . . . . . . . 12 ⊢ (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
5	4	cleq2d 1112	. . . . . . . . . . 11 ⊢ (w = ⟨x, y⟩ → (v = ⟨w, z⟩ ↔ v = ⟨⟨x, y⟩, z⟩))
6	5	pm5.32ri 490	. . . . . . . . . 10 ⊢ ((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩))
7	6	anbi1i 368	. . . . . . . . 9 ⊢ (((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩) ∧ φ))
8		anass 336	. . . . . . . . 9 ⊢ (((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ (v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
9		an23 371	. . . . . . . . 9 ⊢ (((v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
10	7, 8, 9	3bitr3 156	. . . . . . . 8 ⊢ ((v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
11	10	biex 733	. . . . . . 7 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
12		opex 1893	. . . . . . . . 9 ⊢ ⟨x, y⟩ ∈ V
13	12	isseti 1352	. . . . . . . 8 ⊢ ∃w w = ⟨x, y⟩
14	13	biantru 543	. . . . . . 7 ⊢ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ ∃w w = ⟨x, y⟩))
15	3, 11, 14	3bitr4 158	. . . . . 6 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ φ))
16	15	bi3ex 735	. . . . 5 ⊢ (∃x∃y∃z∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
17	2, 16	bitr 151	. . . 4 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
18		19.42vv 968	. . . . 5 ⊢ (∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
19	18	bi2ex 734	. . . 4 ⊢ (∃w∃z∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
20	1, 17, 19	3bitr3 156	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
21	20	biabi 1181	. 2 ⊢ {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} = {v∣∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
22		df-oprab 3004	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {v∣∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
23		df-opab 2098	. 2 ⊢ {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {v∣∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
24	21, 22, 23	3eqtr4 1126	1 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091  ⟨cop 1810  {copab 2055  {copab2 3002

This theorem is referenced by:  reloprab 3022  bioprabd 3025  cbvoprab3v 3030  dmoprab 3031  rnoprab 3033  ssoprab2i 3036  funoprab 3037  fnoprval 3042

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-oprab 3004

metamath.org