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Theorem dmopabss 4575
 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 4574 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
2 19.42v 1802 . . . 4 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
32abbii 2169 . . 3 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
4 ssab2 3052 . . 3 {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ⊆ 𝐴
53, 4eqsstri 3003 . 2 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} ⊆ 𝐴
61, 5eqsstri 3003 1 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
 Colors of variables: wff set class Syntax hints:   ∧ wa 101  ∃wex 1397   ∈ wcel 1409  {cab 2042   ⊆ wss 2945  {copab 3845  dom cdm 4373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-dm 4383 This theorem is referenced by:  opabex  5413
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