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Theorem dmoprabss 5744
Description: The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
dmoprabss dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem dmoprabss
StepHypRef Expression
1 dmoprab 5743 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 19.42v 1835 . . . 4 (∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑))
32opabbii 3911 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)}
4 opabssxp 4525 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵)
53, 4eqsstri 3057 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
61, 5eqsstri 3057 1 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wex 1427  wcel 1439  wss 3000  {copab 3904   × cxp 4449  dom cdm 4451  {coprab 5667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4457  df-dm 4461  df-oprab 5670
This theorem is referenced by:  elmpt2cl  5856  oprabexd  5912  oprabex  5913
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