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Theorem dmoprabss 7520
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 7519 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 19.42v 1949 . . . 4 (∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑))
32opabbii 5210 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)}
4 opabssxp 5764 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵)
53, 4eqsstri 4007 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
61, 5eqsstri 4007 1 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 394  wex 1773  wcel 2098  wss 3939  {copab 5205   × cxp 5670  dom cdm 5672  {coprab 7417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-dm 5682  df-oprab 7420
This theorem is referenced by:  mpondm0  7658  elmpocl  7659  oprabexd  7977  oprabex  7978  bropopvvv  8093  bropfvvvv  8095  dmaddsr  11108  dmmulsr  11109  axaddf  11168  axmulf  11169
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