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Theorem resopab 6035
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 5689 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2 df-xp 5683 . . . . . 6 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
3 vex 3479 . . . . . . . 8 𝑦 ∈ V
43biantru 531 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴𝑦 ∈ V))
54opabbii 5216 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
62, 5eqtr4i 2764 . . . . 5 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}
76ineq2i 4210 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴})
8 incom 4202 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
97, 8eqtri 2761 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10 inopab 5830 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
119, 10eqtri 2761 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
121, 11eqtri 2761 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cin 3948  {copab 5211   × cxp 5675  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by:  resopab2  6037  opabresid  6050  mptpreima  6238  isarep2  6640  resoprab  7526  elrnmpores  7546  df1st2  8084  df2nd2  8085  imaopab  41050
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