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Theorem resopab 5880
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 5544 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2 df-xp 5538 . . . . . 6 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
3 vex 3472 . . . . . . . 8 𝑦 ∈ V
43biantru 533 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴𝑦 ∈ V))
54opabbii 5109 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ V)}
62, 5eqtr4i 2848 . . . . 5 (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}
76ineq2i 4160 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴})
8 incom 4152 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
97, 8eqtri 2845 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10 inopab 5678 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
119, 10eqtri 2845 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
121, 11eqtri 2845 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  cin 3907  {copab 5104   × cxp 5530  cres 5534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-opab 5105  df-xp 5538  df-rel 5539  df-res 5544
This theorem is referenced by:  resopab2  5882  opabresid  5895  opabresidOLD  5897  mptpreima  6070  isarep2  6422  resoprab  7254  elrnmpores  7272  df1st2  7780  df2nd2  7781  imaopab  39360
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