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Theorem resopab2 6056
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
Assertion
Ref Expression
resopab2 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem resopab2
StepHypRef Expression
1 resopab 6054 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))}
2 ssel 3989 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32pm4.71d 561 . . . . 5 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
43anbi1d 631 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑)))
5 anass 468 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
64, 5bitr2di 288 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐴𝜑)))
76opabbidv 5214 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
81, 7eqtrid 2787 1 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963  {copab 5210  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  resmpt  6057  marypha2lem4  9476
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