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Theorem resopab2 5747
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 5745 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))}
2 ssel 3851 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32pm4.71d 554 . . . . 5 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
43anbi1d 620 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑)))
5 anass 461 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
64, 5syl6rbb 280 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐴𝜑)))
76opabbidv 4993 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
81, 7syl5eq 2823 1 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  wss 3828  {copab 4989  cres 5406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pr 5184
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-rab 3094  df-v 3414  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-sn 4440  df-pr 4442  df-op 4446  df-opab 4990  df-xp 5410  df-rel 5411  df-res 5416
This theorem is referenced by:  resmpt  5748  marypha2lem4  8693
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