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Theorem dfoprab2 5822
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 1643 . . . 4 (∃𝑧𝑤𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤𝑧𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2 exrot4 1670 . . . . 5 (∃𝑧𝑤𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦𝑧𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3 opeq1 3709 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
43eqeq2d 2152 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑣 = ⟨𝑤, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54pm5.32ri 451 . . . . . . . . . 10 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
65anbi1i 454 . . . . . . . . 9 (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑))
7 anass 399 . . . . . . . . 9 (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8 an32 552 . . . . . . . . 9 (((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
96, 7, 83bitr3i 209 . . . . . . . 8 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
109exbii 1585 . . . . . . 7 (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
11 vex 2690 . . . . . . . . . 10 𝑥 ∈ V
12 vex 2690 . . . . . . . . . 10 𝑦 ∈ V
1311, 12opex 4155 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
1413isseti 2695 . . . . . . . 8 𝑤 𝑤 = ⟨𝑥, 𝑦
15 19.42v 1879 . . . . . . . 8 (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩))
1614, 15mpbiran2 926 . . . . . . 7 (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
1710, 16bitri 183 . . . . . 6 (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
18173exbii 1587 . . . . 5 (∃𝑥𝑦𝑧𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
192, 18bitri 183 . . . 4 (∃𝑧𝑤𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
20 19.42vv 1884 . . . . 5 (∃𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
21202exbii 1586 . . . 4 (∃𝑤𝑧𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
221, 19, 213bitr3i 209 . . 3 (∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2322abbii 2256 . 2 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
24 df-oprab 5782 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
25 df-opab 3994 . 2 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
2623, 24, 253eqtr4i 2171 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wex 1469  {cab 2126  cop 3531  {copab 3992  {coprab 5779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-opab 3994  df-oprab 5782
This theorem is referenced by:  reloprab  5823  cbvoprab1  5847  cbvoprab12  5849  cbvoprab3  5851  dmoprab  5856  rnoprab  5858  ssoprab2i  5864  mpomptx  5866  resoprab  5871  funoprabg  5874  ov6g  5912  dfoprab3s  6092  xpcomco  6724
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