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Theorem unxpwdom 8441
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 7908 . . . . 5 Rel ≼
21brrelex2i 5121 . . . 4 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐵𝐶) ∈ V)
3 domeng 7916 . . . 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 256 . 2 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → ∃𝑥((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶)))
6 simprl 793 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7 indi 3851 . . . . . 6 (𝑥 ∩ (𝐵𝐶)) = ((𝑥𝐵) ∪ (𝑥𝐶))
8 simprr 795 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝑥 ⊆ (𝐵𝐶))
9 df-ss 3570 . . . . . . 7 (𝑥 ⊆ (𝐵𝐶) ↔ (𝑥 ∩ (𝐵𝐶)) = 𝑥)
108, 9sylib 208 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥 ∩ (𝐵𝐶)) = 𝑥)
117, 10syl5eqr 2669 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → ((𝑥𝐵) ∪ (𝑥𝐶)) = 𝑥)
126, 11breqtrrd 4643 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥𝐵) ∪ (𝑥𝐶)))
13 unxpwdom2 8440 . . . 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 3756 . . . . . . . 8 𝐵 ⊆ (𝐵𝐶)
162adantr 481 . . . . . . . 8 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐵𝐶) ∈ V)
17 ssexg 4766 . . . . . . . 8 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
1815, 16, 17sylancr 694 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐵 ∈ V)
19 inss2 3814 . . . . . . 7 (𝑥𝐵) ⊆ 𝐵
20 ssdomg 7948 . . . . . . 7 (𝐵 ∈ V → ((𝑥𝐵) ⊆ 𝐵 → (𝑥𝐵) ≼ 𝐵))
2118, 19, 20mpisyl 21 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼ 𝐵)
22 domwdom 8426 . . . . . 6 ((𝑥𝐵) ≼ 𝐵 → (𝑥𝐵) ≼* 𝐵)
2321, 22syl 17 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼* 𝐵)
24 wdomtr 8427 . . . . . 6 ((𝐴* (𝑥𝐵) ∧ (𝑥𝐵) ≼* 𝐵) → 𝐴* 𝐵)
2524expcom 451 . . . . 5 ((𝑥𝐵) ≼* 𝐵 → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
2623, 25syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
27 ssun2 3757 . . . . . . 7 𝐶 ⊆ (𝐵𝐶)
28 ssexg 4766 . . . . . . 7 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
2927, 16, 28sylancr 694 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐶 ∈ V)
30 inss2 3814 . . . . . 6 (𝑥𝐶) ⊆ 𝐶
31 ssdomg 7948 . . . . . 6 (𝐶 ∈ V → ((𝑥𝐶) ⊆ 𝐶 → (𝑥𝐶) ≼ 𝐶))
3229, 30, 31mpisyl 21 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐶) ≼ 𝐶)
33 domtr 7956 . . . . . 6 ((𝐴 ≼ (𝑥𝐶) ∧ (𝑥𝐶) ≼ 𝐶) → 𝐴𝐶)
3433expcom 451 . . . . 5 ((𝑥𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3532, 34syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3626, 35orim12d 882 . . 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 1860 1 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cun 3554  cin 3555  wss 3556   class class class wbr 4615   × cxp 5074  cen 7899  cdom 7900  * cwdom 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-1st 7116  df-2nd 7117  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-wdom 8411
This theorem is referenced by:  pwcdadom  8985
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