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Theorem unxpwdom 9586
Description: If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
unxpwdom ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))

Proof of Theorem unxpwdom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 reldom 8947 . . . . 5 Rel ≼
21brrelex2i 5732 . . . 4 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐵𝐶) ∈ V)
3 domeng 8960 . . . 4 ((𝐵𝐶) ∈ V → ((𝐴 × 𝐴) ≼ (𝐵𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))))
42, 3syl 17 . . 3 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → ((𝐴 × 𝐴) ≼ (𝐵𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))))
54ibi 266 . 2 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → ∃𝑥((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶)))
6 simprl 767 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7 indi 4272 . . . . . 6 (𝑥 ∩ (𝐵𝐶)) = ((𝑥𝐵) ∪ (𝑥𝐶))
8 simprr 769 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝑥 ⊆ (𝐵𝐶))
9 df-ss 3964 . . . . . . 7 (𝑥 ⊆ (𝐵𝐶) ↔ (𝑥 ∩ (𝐵𝐶)) = 𝑥)
108, 9sylib 217 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥 ∩ (𝐵𝐶)) = 𝑥)
117, 10eqtr3id 2784 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → ((𝑥𝐵) ∪ (𝑥𝐶)) = 𝑥)
126, 11breqtrrd 5175 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥𝐵) ∪ (𝑥𝐶)))
13 unxpwdom2 9585 . . . 4 ((𝐴 × 𝐴) ≈ ((𝑥𝐵) ∪ (𝑥𝐶)) → (𝐴* (𝑥𝐵) ∨ 𝐴 ≼ (𝑥𝐶)))
1412, 13syl 17 . . 3 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴* (𝑥𝐵) ∨ 𝐴 ≼ (𝑥𝐶)))
15 ssun1 4171 . . . . . . . 8 𝐵 ⊆ (𝐵𝐶)
162adantr 479 . . . . . . . 8 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐵𝐶) ∈ V)
17 ssexg 5322 . . . . . . . 8 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
1815, 16, 17sylancr 585 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐵 ∈ V)
19 inss2 4228 . . . . . . 7 (𝑥𝐵) ⊆ 𝐵
20 ssdomg 8998 . . . . . . 7 (𝐵 ∈ V → ((𝑥𝐵) ⊆ 𝐵 → (𝑥𝐵) ≼ 𝐵))
2118, 19, 20mpisyl 21 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼ 𝐵)
22 domwdom 9571 . . . . . 6 ((𝑥𝐵) ≼ 𝐵 → (𝑥𝐵) ≼* 𝐵)
2321, 22syl 17 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼* 𝐵)
24 wdomtr 9572 . . . . . 6 ((𝐴* (𝑥𝐵) ∧ (𝑥𝐵) ≼* 𝐵) → 𝐴* 𝐵)
2524expcom 412 . . . . 5 ((𝑥𝐵) ≼* 𝐵 → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
2623, 25syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
27 ssun2 4172 . . . . . . 7 𝐶 ⊆ (𝐵𝐶)
28 ssexg 5322 . . . . . . 7 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
2927, 16, 28sylancr 585 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐶 ∈ V)
30 inss2 4228 . . . . . 6 (𝑥𝐶) ⊆ 𝐶
31 ssdomg 8998 . . . . . 6 (𝐶 ∈ V → ((𝑥𝐶) ⊆ 𝐶 → (𝑥𝐶) ≼ 𝐶))
3229, 30, 31mpisyl 21 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐶) ≼ 𝐶)
33 domtr 9005 . . . . . 6 ((𝐴 ≼ (𝑥𝐶) ∧ (𝑥𝐶) ≼ 𝐶) → 𝐴𝐶)
3433expcom 412 . . . . 5 ((𝑥𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3532, 34syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3626, 35orim12d 961 . . 3 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → ((𝐴* (𝑥𝐵) ∨ 𝐴 ≼ (𝑥𝐶)) → (𝐴* 𝐵𝐴𝐶)))
3714, 36mpd 15 . 2 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴* 𝐵𝐴𝐶))
385, 37exlimddv 1936 1 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 843   = wceq 1539  wex 1779  wcel 2104  Vcvv 3472  cun 3945  cin 3946  wss 3947   class class class wbr 5147   × cxp 5673  cen 8938  cdom 8939  * cwdom 9561
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7977  df-2nd 7978  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-wdom 9562
This theorem is referenced by:  pwdjudom  10213
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