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Theorem resopab 4680
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 4385 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2 df-xp 4379 . . . . . 6 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
3 vex 2577 . . . . . . . 8 𝑦 ∈ V
43biantru 290 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴𝑦 ∈ V))
54opabbii 3852 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
62, 5eqtr4i 2079 . . . . 5 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}
76ineq2i 3163 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴})
8 incom 3157 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
97, 8eqtri 2076 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10 inopab 4496 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
119, 10eqtri 2076 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
121, 11eqtri 2076 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cin 2944  {copab 3845   × cxp 4371  cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by:  resopab2  4683  opabresid  4687  mptpreima  4842  isarep2  5014  resoprab  5625  df1st2  5868  df2nd2  5869
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