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Theorem resopab 5481
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
Assertion
Ref Expression
resopab ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem resopab
StepHypRef Expression
1 df-res 5155 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2 df-xp 5149 . . . . . 6 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
3 vex 3234 . . . . . . . 8 𝑦 ∈ V
43biantru 525 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴𝑦 ∈ V))
54opabbii 4750 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
62, 5eqtr4i 2676 . . . . 5 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}
76ineq2i 3844 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴})
8 incom 3838 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
97, 8eqtri 2673 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10 inopab 5285 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
119, 10eqtri 2673 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
121, 11eqtri 2673 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  {copab 4745   × cxp 5141  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-res 5155
This theorem is referenced by:  resopab2  5483  opabresid  5490  mptpreima  5666  isarep2  6016  resoprab  6798  elrnmpt2res  6816  df1st2  7308  df2nd2  7309
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