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Theorem difxp 5593
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 3770 . . 3 ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ⊆ (𝐶 × 𝐷)
2 relxp 5160 . . 3 Rel (𝐶 × 𝐷)
3 relss 5240 . . 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 5160 . . 3 Rel ((𝐶𝐴) × 𝐷)
6 relxp 5160 . . 3 Rel (𝐶 × (𝐷𝐵))
7 relun 5268 . . 3 (Rel (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵))) ↔ (Rel ((𝐶𝐴) × 𝐷) ∧ Rel (𝐶 × (𝐷𝐵))))
85, 6, 7mpbir2an 975 . 2 Rel (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵)))
9 ianor 508 . . . . . 6 (¬ (𝑥𝐴𝑦𝐵) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝑦𝐵))
109anbi2i 730 . . . . 5 (((𝑥𝐶𝑦𝐷) ∧ ¬ (𝑥𝐴𝑦𝐵)) ↔ ((𝑥𝐶𝑦𝐷) ∧ (¬ 𝑥𝐴 ∨ ¬ 𝑦𝐵)))
11 andi 929 . . . . 5 (((𝑥𝐶𝑦𝐷) ∧ (¬ 𝑥𝐴 ∨ ¬ 𝑦𝐵)) ↔ (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴) ∨ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵)))
1210, 11bitri 264 . . . 4 (((𝑥𝐶𝑦𝐷) ∧ ¬ (𝑥𝐴𝑦𝐵)) ↔ (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴) ∨ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵)))
13 opelxp 5180 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥𝐶𝑦𝐷))
14 opelxp 5180 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1514notbii 309 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ ¬ (𝑥𝐴𝑦𝐵))
1613, 15anbi12i 733 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ (𝑥𝐴𝑦𝐵)))
17 opelxp 5180 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ↔ (𝑥 ∈ (𝐶𝐴) ∧ 𝑦𝐷))
18 eldif 3617 . . . . . . . 8 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
1918anbi1i 731 . . . . . . 7 ((𝑥 ∈ (𝐶𝐴) ∧ 𝑦𝐷) ↔ ((𝑥𝐶 ∧ ¬ 𝑥𝐴) ∧ 𝑦𝐷))
20 an32 856 . . . . . . 7 (((𝑥𝐶 ∧ ¬ 𝑥𝐴) ∧ 𝑦𝐷) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴))
2119, 20bitri 264 . . . . . 6 ((𝑥 ∈ (𝐶𝐴) ∧ 𝑦𝐷) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴))
2217, 21bitri 264 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴))
23 eldif 3617 . . . . . . 7 (𝑦 ∈ (𝐷𝐵) ↔ (𝑦𝐷 ∧ ¬ 𝑦𝐵))
2423anbi2i 730 . . . . . 6 ((𝑥𝐶𝑦 ∈ (𝐷𝐵)) ↔ (𝑥𝐶 ∧ (𝑦𝐷 ∧ ¬ 𝑦𝐵)))
25 opelxp 5180 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵)) ↔ (𝑥𝐶𝑦 ∈ (𝐷𝐵)))
26 anass 682 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵) ↔ (𝑥𝐶 ∧ (𝑦𝐷 ∧ ¬ 𝑦𝐵)))
2724, 25, 263bitr4i 292 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵)) ↔ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵))
2822, 27orbi12i 542 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵))) ↔ (((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑥𝐴) ∨ ((𝑥𝐶𝑦𝐷) ∧ ¬ 𝑦𝐵)))
2912, 16, 283bitr4i 292 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵))))
30 eldif 3617 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
31 elun 3786 . . 3 (⟨𝑥, 𝑦⟩ ∈ (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ ((𝐶𝐴) × 𝐷) ∨ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × (𝐷𝐵))))
3229, 30, 313bitr4i 292 . 2 (⟨𝑥, 𝑦⟩ ∈ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵))))
334, 8, 32eqrelriiv 5248 1 ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶𝐴) × 𝐷) ∪ (𝐶 × (𝐷𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1523  wcel 2030  cdif 3604  cun 3605  wss 3607  cop 4216   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  difxp1  5594  difxp2  5595  evlslem4  19556  txcld  21454
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