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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⟩ ∣ 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 1536 . . . 4 (zwxy(v = ⟨w, z (w = ⟨x, y φ)) ↔ wzxy(v = ⟨w, z (w = ⟨x, y φ)))
2 exrot4 1563 . . . . 5 (zwxy(v = ⟨w, z (w = ⟨x, y φ)) ↔ xyzw(v = ⟨w, z (w = ⟨x, y φ)))
3 opeq1 3523 . . . . . . . . . . . 12 (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
43eqeq2d 2033 . . . . . . . . . . 11 (w = ⟨x, y⟩ → (v = ⟨w, z⟩ ↔ v = ⟨⟨x, y⟩, z⟩))
54pm5.32ri 431 . . . . . . . . . 10 ((v = ⟨w, z w = ⟨x, y⟩) ↔ (v = ⟨⟨x, y⟩, z w = ⟨x, y⟩))
65anbi1i 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⟩))
96, 7, 83bitr3i 199 . . . . . . . 8 ((v = ⟨w, z (w = ⟨x, y φ)) ↔ ((v = ⟨⟨x, y⟩, z φ) w = ⟨x, y⟩))
109exbii 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
1311, 12opex 3940 . . . . . . . . 9 x, y V
1413isseti 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⟩))
1614, 15mpbiran2 836 . . . . . . 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 1480 . . . . 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 1770 . . . . 5 (xy(v = ⟨w, z (w = ⟨x, y φ)) ↔ (v = ⟨w, z xy(w = ⟨x, y φ)))
21202exbii 1479 . . . 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 2135 . 2 {vxyz(v = ⟨⟨x, y⟩, z φ)} = {vwz(v = ⟨w, z xy(w = ⟨x, y φ))}
24 df-oprab 5440 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {vxyz(v = ⟨⟨x, y⟩, z φ)}
25 df-opab 3793 . 2 {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} = {vwz(v = ⟨w, z xy(w = ⟨x, y φ))}
2623, 24, 253eqtr4i 2052 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
Colors of variables: wff set class
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
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