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Theorem difxp 5775
Description: Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
difxp ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵)))

Proof of Theorem difxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difss 3935 . . 3 ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ⊆ (𝐶 × 𝐷)
2 relxp 5330 . . 3 Rel (𝐶 × 𝐷)
3 relss 5411 . . 3 (((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ⊆ (𝐶 × 𝐷) → (Rel (𝐶 × 𝐷) → Rel ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵))))
41, 2, 3mp2 9 . 2 Rel ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵))
5 relxp 5330 . . 3 Rel ((𝐶𝐴) × 𝐷)
6 relxp 5330 . . 3 Rel (𝐶 × (𝐷𝐵))
7 relun 5438 . . 3 (Rel (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵))) ↔ (Rel ((𝐶𝐴) × 𝐷) ∧ Rel (𝐶 × (𝐷𝐵))))
85, 6, 7mpbir2an 703 . 2 Rel (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵)))
9 ianor 1005 . . . . . 6 (¬ (𝑥𝐴𝑦𝐵) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝑦𝐵))
109anbi2i 617 . . . . 5 (((𝑥𝐶𝑦𝐷) ∧ ¬ (𝑥𝐴𝑦𝐵)) ↔ ((𝑥𝐶𝑦𝐷) ∧ (¬ 𝑥𝐴 ∨ ¬ 𝑦𝐵)))
11 andi 1031 . . . . 5 (((𝑥𝐶𝑦𝐷) ∧ (¬ 𝑥𝐴 ∨ ¬ 𝑦𝐵)) ↔ (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴) ∨ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵)))
1210, 11bitri 267 . . . 4 (((𝑥𝐶𝑦𝐷) ∧ ¬ (𝑥𝐴𝑦𝐵)) ↔ (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴) ∨ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵)))
13 opelxp 5348 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥𝐶𝑦𝐷))
14 opelxp 5348 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1514notbii 312 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ ¬ (𝑥𝐴𝑦𝐵))
1613, 15anbi12i 621 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ (𝑥𝐴𝑦𝐵)))
17 opelxp 5348 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ↔ (𝑥 ∈ (𝐶𝐴) ∧ 𝑦𝐷))
18 eldif 3779 . . . . . . . 8 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1918anbi1i 618 . . . . . . 7 ((𝑥 ∈ (𝐶𝐴) ∧ 𝑦𝐷) ↔ ((𝑥𝐶 ∧ ¬ 𝑥𝐴) ∧ 𝑦𝐷))
20 an32 637 . . . . . . 7 (((𝑥𝐶 ∧ ¬ 𝑥𝐴) ∧ 𝑦𝐷) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴))
2119, 20bitri 267 . . . . . 6 ((𝑥 ∈ (𝐶𝐴) ∧ 𝑦𝐷) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴))
2217, 21bitri 267 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴))
23 eldif 3779 . . . . . . 7 (𝑦 ∈ (𝐷𝐵) ↔ (𝑦𝐷 ∧ ¬ 𝑦𝐵))
2423anbi2i 617 . . . . . 6 ((𝑥𝐶𝑦 ∈ (𝐷𝐵)) ↔ (𝑥𝐶 ∧ (𝑦𝐷 ∧ ¬ 𝑦𝐵)))
25 opelxp 5348 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵)) ↔ (𝑥𝐶𝑦 ∈ (𝐷𝐵)))
26 anass 461 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵) ↔ (𝑥𝐶 ∧ (𝑦𝐷 ∧ ¬ 𝑦𝐵)))
2724, 25, 263bitr4i 295 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵)) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵))
2822, 27orbi12i 939 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵))) ↔ (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴) ∨ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵)))
2912, 16, 283bitr4i 295 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵))))
30 eldif 3779 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
31 elun 3951 . . 3 (⟨𝑥, 𝑦⟩ ∈ (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵))))
3229, 30, 313bitr4i 295 . 2 (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵))))
334, 8, 32eqrelriiv 5418 1 ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 385  wo 874   = wceq 1653  wcel 2157  cdif 3766  cun 3767  wss 3769  cop 4374   × cxp 5310  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-xp 5318  df-rel 5319
This theorem is referenced by:  difxp1  5776  difxp2  5777  evlslem4  19830  txcld  21735
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