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Theorem resopab2 4655
 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 4652 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))}
2 ssel 2939 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
32pm4.71d 373 . . . . 5 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
43anbi1d 438 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑)))
5 anass 381 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
64, 5syl6rbb 186 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ (𝑥𝐵𝜑)) ↔ (𝑥𝐴𝜑)))
76opabbidv 3823 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴 ∧ (𝑥𝐵𝜑))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
81, 7syl5eq 2084 1 (𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  {copab 3817   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  resmpt  4656
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