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Theorem resoprab2 6655
Description: Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
resoprab2 ((𝐶𝐴𝐷𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem resoprab2
StepHypRef Expression
1 resoprab 6654 . 2 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))}
2 anass 679 . . . 4 ((((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝜑) ↔ ((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
3 an4 861 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ↔ ((𝑥𝐶𝑥𝐴) ∧ (𝑦𝐷𝑦𝐵)))
4 ssel 3562 . . . . . . . . 9 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
54pm4.71d 664 . . . . . . . 8 (𝐶𝐴 → (𝑥𝐶 ↔ (𝑥𝐶𝑥𝐴)))
65bicomd 212 . . . . . . 7 (𝐶𝐴 → ((𝑥𝐶𝑥𝐴) ↔ 𝑥𝐶))
7 ssel 3562 . . . . . . . . 9 (𝐷𝐵 → (𝑦𝐷𝑦𝐵))
87pm4.71d 664 . . . . . . . 8 (𝐷𝐵 → (𝑦𝐷 ↔ (𝑦𝐷𝑦𝐵)))
98bicomd 212 . . . . . . 7 (𝐷𝐵 → ((𝑦𝐷𝑦𝐵) ↔ 𝑦𝐷))
106, 9bi2anan9 913 . . . . . 6 ((𝐶𝐴𝐷𝐵) → (((𝑥𝐶𝑥𝐴) ∧ (𝑦𝐷𝑦𝐵)) ↔ (𝑥𝐶𝑦𝐷)))
113, 10syl5bb 271 . . . . 5 ((𝐶𝐴𝐷𝐵) → (((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑥𝐶𝑦𝐷)))
1211anbi1d 737 . . . 4 ((𝐶𝐴𝐷𝐵) → ((((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝜑) ↔ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)))
132, 12syl5bbr 273 . . 3 ((𝐶𝐴𝐷𝐵) → (((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)) ↔ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)))
1413oprabbidv 6607 . 2 ((𝐶𝐴𝐷𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
151, 14syl5eq 2656 1 ((𝐶𝐴𝐷𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wss 3540   × cxp 5036  cres 5040  {coprab 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-res 5050  df-oprab 6553
This theorem is referenced by:  resmpt2  6656
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