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Theorem resopab2 4759
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 4756 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))}
2 ssel 3019 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32pm4.71d 385 . . . . 5 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
43anbi1d 453 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑)))
5 anass 393 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
64, 5syl6rbb 195 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐴𝜑)))
76opabbidv 3904 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
81, 7syl5eq 2132 1 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wss 2999  {copab 3898  cres 4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445  df-res 4450
This theorem is referenced by:  resmpt  4760
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