Theorem dfoprab2 5494

Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)

Assertion

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

Distinct variable groups:   x,z,w   y,z,w   φ,w

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

Proof of Theorem dfoprab2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 1551	. . . 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 1578	. . . . 5 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
3		opeq1 3540	. . . . . . . . . . . 12 ⊢ (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
4	3	eqeq2d 2048	. . . . . . . . . . 11 ⊢ (w = ⟨x, y⟩ → (v = ⟨w, z⟩ ↔ v = ⟨⟨x, y⟩, z⟩))
5	4	pm5.32ri 428	. . . . . . . . . 10 ⊢ ((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩))
6	5	anbi1i 431	. . . . . . . . 9 ⊢ (((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩) ∧ φ))
7		anass 381	. . . . . . . . 9 ⊢ (((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ (v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
8		an32 496	. . . . . . . . 9 ⊢ (((v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
9	6, 7, 8	3bitr3i 199	. . . . . . . 8 ⊢ ((v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
10	9	exbii 1493	. . . . . . 7 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
11		vex 2554	. . . . . . . . . 10 ⊢ x ∈ V
12		vex 2554	. . . . . . . . . 10 ⊢ y ∈ V
13	11, 12	opex 3957	. . . . . . . . 9 ⊢ ⟨x, y⟩ ∈ V
14	13	isseti 2557	. . . . . . . 8 ⊢ ∃w w = ⟨x, y⟩
15		19.42v 1783	. . . . . . . 8 ⊢ (∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ ∃w w = ⟨x, y⟩))
16	14, 15	mpbiran2 847	. . . . . . 7 ⊢ (∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ φ))
17	10, 16	bitri 173	. . . . . 6 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ φ))
18	17	3exbii 1495	. . . . 5 ⊢ (∃x∃y∃z∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
19	2, 18	bitri 173	. . . 4 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
20		19.42vv 1785	. . . . 5 ⊢ (∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
21	20	2exbii 1494	. . . 4 ⊢ (∃w∃z∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
22	1, 19, 21	3bitr3i 199	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
23	22	abbii 2150	. 2 ⊢ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} = {v ∣ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
24		df-oprab 5459	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
25		df-opab 3810	. 2 ⊢ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {v ∣ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
26	23, 24, 25	3eqtr4i 2067	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378  {cab 2023  ⟨cop 3370  {copab 3808  {coprab 5456

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-oprab 5459

This theorem is referenced by:  reloprab  5495  cbvoprab1  5518  cbvoprab12  5520  cbvoprab3  5522  dmoprab  5527  rnoprab  5529  ssoprab2i  5535  mpt2mptx  5537  resoprab  5539  funoprabg  5542  ov6g  5580  dfoprab3s  5758  xpcomco  6236