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Theorem resoprab2 7274
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 7273 . 2 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))}
2 anass 471 . . . 4 ((((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝜑) ↔ ((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)))
3 an4 654 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ↔ ((𝑥𝐶𝑥𝐴) ∧ (𝑦𝐷𝑦𝐵)))
4 ssel 3964 . . . . . . . . 9 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
54pm4.71d 564 . . . . . . . 8 (𝐶𝐴 → (𝑥𝐶 ↔ (𝑥𝐶𝑥𝐴)))
65bicomd 225 . . . . . . 7 (𝐶𝐴 → ((𝑥𝐶𝑥𝐴) ↔ 𝑥𝐶))
7 ssel 3964 . . . . . . . . 9 (𝐷𝐵 → (𝑦𝐷𝑦𝐵))
87pm4.71d 564 . . . . . . . 8 (𝐷𝐵 → (𝑦𝐷 ↔ (𝑦𝐷𝑦𝐵)))
98bicomd 225 . . . . . . 7 (𝐷𝐵 → ((𝑦𝐷𝑦𝐵) ↔ 𝑦𝐷))
106, 9bi2anan9 637 . . . . . 6 ((𝐶𝐴𝐷𝐵) → (((𝑥𝐶𝑥𝐴) ∧ (𝑦𝐷𝑦𝐵)) ↔ (𝑥𝐶𝑦𝐷)))
113, 10syl5bb 285 . . . . 5 ((𝐶𝐴𝐷𝐵) → (((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑥𝐶𝑦𝐷)))
1211anbi1d 631 . . . 4 ((𝐶𝐴𝐷𝐵) → ((((𝑥𝐶𝑦𝐷) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝜑) ↔ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)))
132, 12syl5bbr 287 . . 3 ((𝐶𝐴𝐷𝐵) → (((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)) ↔ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)))
1413oprabbidv 7223 . 2 ((𝐶𝐴𝐷𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝜑))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
151, 14syl5eq 2871 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 398   = wceq 1536  wcel 2113  wss 3939   × cxp 5556  cres 5560  {coprab 7160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-opab 5132  df-xp 5564  df-rel 5565  df-res 5570  df-oprab 7163
This theorem is referenced by:  resmpo  7275
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