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Theorem unxpwdom 9629
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 8991 . . . . 5 Rel ≼
21brrelex2i 5742 . . . 4 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐵𝐶) ∈ V)
3 domeng 9003 . . . 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 267 . 2 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → ∃𝑥((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶)))
6 simprl 771 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7 indi 4284 . . . . . 6 (𝑥 ∩ (𝐵𝐶)) = ((𝑥𝐵) ∪ (𝑥𝐶))
8 simprr 773 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝑥 ⊆ (𝐵𝐶))
9 dfss2 3969 . . . . . . 7 (𝑥 ⊆ (𝐵𝐶) ↔ (𝑥 ∩ (𝐵𝐶)) = 𝑥)
108, 9sylib 218 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥 ∩ (𝐵𝐶)) = 𝑥)
117, 10eqtr3id 2791 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → ((𝑥𝐵) ∪ (𝑥𝐶)) = 𝑥)
126, 11breqtrrd 5171 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥𝐵) ∪ (𝑥𝐶)))
13 unxpwdom2 9628 . . . 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 4178 . . . . . . . 8 𝐵 ⊆ (𝐵𝐶)
162adantr 480 . . . . . . . 8 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐵𝐶) ∈ V)
17 ssexg 5323 . . . . . . . 8 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
1815, 16, 17sylancr 587 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐵 ∈ V)
19 inss2 4238 . . . . . . 7 (𝑥𝐵) ⊆ 𝐵
20 ssdomg 9040 . . . . . . 7 (𝐵 ∈ V → ((𝑥𝐵) ⊆ 𝐵 → (𝑥𝐵) ≼ 𝐵))
2118, 19, 20mpisyl 21 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼ 𝐵)
22 domwdom 9614 . . . . . 6 ((𝑥𝐵) ≼ 𝐵 → (𝑥𝐵) ≼* 𝐵)
2321, 22syl 17 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼* 𝐵)
24 wdomtr 9615 . . . . . 6 ((𝐴* (𝑥𝐵) ∧ (𝑥𝐵) ≼* 𝐵) → 𝐴* 𝐵)
2524expcom 413 . . . . 5 ((𝑥𝐵) ≼* 𝐵 → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
2623, 25syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
27 ssun2 4179 . . . . . . 7 𝐶 ⊆ (𝐵𝐶)
28 ssexg 5323 . . . . . . 7 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
2927, 16, 28sylancr 587 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐶 ∈ V)
30 inss2 4238 . . . . . 6 (𝑥𝐶) ⊆ 𝐶
31 ssdomg 9040 . . . . . 6 (𝐶 ∈ V → ((𝑥𝐶) ⊆ 𝐶 → (𝑥𝐶) ≼ 𝐶))
3229, 30, 31mpisyl 21 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐶) ≼ 𝐶)
33 domtr 9047 . . . . . 6 ((𝐴 ≼ (𝑥𝐶) ∧ (𝑥𝐶) ≼ 𝐶) → 𝐴𝐶)
3433expcom 413 . . . . 5 ((𝑥𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3532, 34syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3626, 35orim12d 967 . . 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 1935 1 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cun 3949  cin 3950  wss 3951   class class class wbr 5143   × cxp 5683  cen 8982  cdom 8983  * cwdom 9604
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-wdom 9605
This theorem is referenced by:  pwdjudom  10255
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