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Theorem resopab2 4972
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
Assertion
Ref Expression
resopab2 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem resopab2
StepHypRef Expression
1 resopab 4969 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))}
2 ssel 3164 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32pm4.71d 393 . . . . 5 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
43anbi1d 465 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑)))
5 anass 401 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
64, 5bitr2di 197 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐴𝜑)))
76opabbidv 4084 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
81, 7eqtrid 2234 1 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wss 3144  {copab 4078  cres 4646
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-xp 4650  df-rel 4651  df-res 4656
This theorem is referenced by:  resmpt  4973
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