Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmoprabss Structured version   Visualization version   GIF version

Theorem dmoprabss 6784
 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 6783 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 19.42v 1921 . . . 4 (∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑))
32opabbii 4750 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)}
4 opabssxp 5227 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵)
53, 4eqsstri 3668 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
61, 5eqsstri 3668 1 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383  ∃wex 1744   ∈ wcel 2030   ⊆ wss 3607  {copab 4745   × cxp 5141  dom cdm 5143  {coprab 6691 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-oprab 6694 This theorem is referenced by:  mpt2ndm0  6917  elmpt2cl  6918  oprabexd  7197  oprabex  7198  bropopvvv  7300  bropfvvvv  7302  dmaddsr  9944  dmmulsr  9945  axaddf  10004  axmulf  10005
 Copyright terms: Public domain W3C validator