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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⟩ ∣ xy(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.
StepHypRef Expression
1 excom 1551 . . . 4 (zwxy(v = ⟨w, z (w = ⟨x, y φ)) ↔ wzxy(v = ⟨w, z (w = ⟨x, y φ)))
2 exrot4 1578 . . . . 5 (zwxy(v = ⟨w, z (w = ⟨x, y φ)) ↔ xyzw(v = ⟨w, z (w = ⟨x, y φ)))
3 opeq1 3540 . . . . . . . . . . . 12 (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
43eqeq2d 2048 . . . . . . . . . . 11 (w = ⟨x, y⟩ → (v = ⟨w, z⟩ ↔ v = ⟨⟨x, y⟩, z⟩))
54pm5.32ri 428 . . . . . . . . . 10 ((v = ⟨w, z w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z w = ⟨x, y⟩))
65anbi1i 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⟩))
96, 7, 83bitr3i 199 . . . . . . . 8 ((v = ⟨w, z (w = ⟨x, y φ)) ↔ ((v = ⟨⟨x, y⟩, z φ) w = ⟨x, y⟩))
109exbii 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
1311, 12opex 3957 . . . . . . . . 9 x, y V
1413isseti 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⟩))
1614, 15mpbiran2 847 . . . . . . 7 (w((v = ⟨⟨x, y⟩, z φ) w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z φ))
1710, 16bitri 173 . . . . . 6 (w(v = ⟨w, z (w = ⟨x, y φ)) ↔ (v = ⟨⟨x, y⟩, z φ))
18173exbii 1495 . . . . 5 (xyzw(v = ⟨w, z (w = ⟨x, y φ)) ↔ xyz(v = ⟨⟨x, y⟩, z φ))
192, 18bitri 173 . . . 4 (zwxy(v = ⟨w, z (w = ⟨x, y φ)) ↔ xyz(v = ⟨⟨x, y⟩, z φ))
20 19.42vv 1785 . . . . 5 (xy(v = ⟨w, z (w = ⟨x, y φ)) ↔ (v = ⟨w, z xy(w = ⟨x, y φ)))
21202exbii 1494 . . . 4 (wzxy(v = ⟨w, z (w = ⟨x, y φ)) ↔ wz(v = ⟨w, z xy(w = ⟨x, y φ)))
221, 19, 213bitr3i 199 . . 3 (xyz(v = ⟨⟨x, y⟩, z φ) ↔ wz(v = ⟨w, z xy(w = ⟨x, y φ)))
2322abbii 2150 . 2 {vxyz(v = ⟨⟨x, y⟩, z φ)} = {vwz(v = ⟨w, z xy(w = ⟨x, y φ))}
24 df-oprab 5459 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {vxyz(v = ⟨⟨x, y⟩, z φ)}
25 df-opab 3810 . 2 {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} = {vwz(v = ⟨w, z xy(w = ⟨x, y φ))}
2623, 24, 253eqtr4i 2067 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
Colors of variables: wff set class
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
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