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Theorem dmopabss 5774
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 5771 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
2 19.42v 1955 . . . 4 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
32abbii 2889 . . 3 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
4 ssab2 4040 . . 3 {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ⊆ 𝐴
53, 4eqsstri 3986 . 2 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} ⊆ 𝐴
61, 5eqsstri 3986 1 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 399  wex 1781  wcel 2115  {cab 2802  wss 3919  {copab 5114  dom cdm 5542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-dm 5552
This theorem is referenced by:  fvopab4ndm  6785  opabex  6971  perpln1  26500  dmadjss  29666  abrexdomjm  30270  abrexdom  35078
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