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Theorem dmopabss 5875
Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
Assertion
Ref Expression
dmopabss dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dmopabss
StepHypRef Expression
1 dmopab 5872 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
2 19.42v 1958 . . . 4 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
32abbii 2803 . . 3 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
4 ssab2 4037 . . 3 {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ⊆ 𝐴
53, 4eqsstri 3979 . 2 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} ⊆ 𝐴
61, 5eqsstri 3979 1 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397  wex 1782  wcel 2107  {cab 2710  wss 3911  {copab 5168  dom cdm 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-dm 5644
This theorem is referenced by:  fvopab4ndm  6978  opabex  7171  pwfir  9123  perpln1  27694  dmadjss  30871  abrexdomjm  31476  abrexdom  36235
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