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Theorem resopab 4946
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 4634 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2 df-xp 4628 . . . . . 6 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
3 vex 2740 . . . . . . . 8 𝑦 ∈ V
43biantru 302 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴𝑦 ∈ V))
54opabbii 4067 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
62, 5eqtr4i 2201 . . . . 5 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}
76ineq2i 3333 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴})
8 incom 3327 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
97, 8eqtri 2198 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10 inopab 4754 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
119, 10eqtri 2198 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
121, 11eqtri 2198 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cin 3128  {copab 4060   × cxp 4620  cres 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4628  df-rel 4629  df-res 4634
This theorem is referenced by:  resopab2  4949  opabresid  4955  mptpreima  5117  isarep2  5298  resoprab  5964  df1st2  6213  df2nd2  6214
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