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Theorem resoprab 5676
Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
Assertion
Ref Expression
resoprab ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem resoprab
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 resopab 4713 . . 3 ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
2 19.42vv 1831 . . . . 5 (∃𝑥𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3 an12 526 . . . . . . 7 ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)))
4 eleq1 2145 . . . . . . . . . 10 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
5 opelxp 4430 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
64, 5syl6bb 194 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
76anbi1d 453 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
87pm5.32i 442 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
93, 8bitri 182 . . . . . 6 ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
1092exbii 1538 . . . . 5 (∃𝑥𝑦(𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
112, 10bitr3i 184 . . . 4 ((𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
1211opabbii 3871 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))}
131, 12eqtri 2103 . 2 ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵)) = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))}
14 dfoprab2 5631 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
1514reseq1i 4667 . 2 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = ({⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↾ (𝐴 × 𝐵))
16 dfoprab2 5631 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))}
1713, 15, 163eqtr4i 2113 1 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  cop 3425  {copab 3864   × cxp 4399  cres 4403  {coprab 5592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4407  df-rel 4408  df-res 4413  df-oprab 5595
This theorem is referenced by:  resoprab2  5677
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