Theorem dfoprab2 5475

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 1536	. . . 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 1563	. . . . 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 3523	. . . . . . . . . . . 12 ⊢ (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
4	3	eqeq2d 2033	. . . . . . . . . . 11 ⊢ (w = ⟨x, y⟩ → (v = ⟨w, z⟩ ↔ v = ⟨⟨x, y⟩, z⟩))
5	4	pm5.32ri 431	. . . . . . . . . 10 ⊢ ((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩))
6	5	anbi1i 434	. . . . . . . . 9 ⊢ (((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ w = ⟨x, y⟩) ∧ φ))
7		anass 383	. . . . . . . . 9 ⊢ (((v = ⟨w, z⟩ ∧ w = ⟨x, y⟩) ∧ φ) ↔ (v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
8		an32 484	. . . . . . . . 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 1478	. . . . . . 7 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩))
11		vex 2538	. . . . . . . . . 10 ⊢ x ∈ V
12		vex 2538	. . . . . . . . . 10 ⊢ y ∈ V
13	11, 12	opex 3940	. . . . . . . . 9 ⊢ ⟨x, y⟩ ∈ V
14	13	isseti 2541	. . . . . . . 8 ⊢ ∃w w = ⟨x, y⟩
15		19.42v 1768	. . . . . . . 8 ⊢ (∃w((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ w = ⟨x, y⟩) ↔ ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ∧ ∃w w = ⟨x, y⟩))
16	14, 15	mpbiran2 836	. . . . . . 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 1480	. . . . 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 1770	. . . . 5 ⊢ (∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
21	20	2exbii 1479	. . . 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 2135	. 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 5440	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
25		df-opab 3793	. 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 2052	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}

Syntax hints:   ∧ wa 97   = wceq 1228  ∃wex 1362  {cab 2008  ⟨cop 3353  {copab 3791  {coprab 5437

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793  df-oprab 5440

This theorem is referenced by:  reloprab  5476  cbvoprab1  5499  cbvoprab12  5501  cbvoprab3  5503  dmoprab  5508  rnoprab  5510  ssoprab2i  5516  mpt2mptx  5518  resoprab  5520  funoprabg  5523  ov6g  5561  dfoprab3s  5739