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Theorem dmoprabss 6640
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 6639 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 19.42v 1905 . . . 4 (∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑))
32opabbii 4649 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)}
4 opabssxp 5116 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵)
53, 4eqsstri 3598 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
61, 5eqsstri 3598 1 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1695  wcel 1977  wss 3540  {copab 4642   × cxp 5036  dom cdm 5038  {coprab 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-oprab 6553
This theorem is referenced by:  mpt2ndm0  6773  elmpt2cl  6774  oprabexd  7046  oprabex  7047  bropopvvv  7142  bropfvvvv  7144  dmaddsr  9785  dmmulsr  9786  axaddf  9845  axmulf  9846  2wlkonot3v  26402  2spthonot3v  26403
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