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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 5726 . . . 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 267 . 2 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → ∃𝑥((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶)))
6 simprl 768 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7 indi 4268 . . . . . 6 (𝑥 ∩ (𝐵𝐶)) = ((𝑥𝐵) ∪ (𝑥𝐶))
8 simprr 770 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝑥 ⊆ (𝐵𝐶))
9 df-ss 3960 . . . . . . 7 (𝑥 ⊆ (𝐵𝐶) ↔ (𝑥 ∩ (𝐵𝐶)) = 𝑥)
108, 9sylib 217 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥 ∩ (𝐵𝐶)) = 𝑥)
117, 10eqtr3id 2780 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → ((𝑥𝐵) ∪ (𝑥𝐶)) = 𝑥)
126, 11breqtrrd 5169 . . . 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 4167 . . . . . . . 8 𝐵 ⊆ (𝐵𝐶)
162adantr 480 . . . . . . . 8 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐵𝐶) ∈ V)
17 ssexg 5316 . . . . . . . 8 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
1815, 16, 17sylancr 586 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐵 ∈ V)
19 inss2 4224 . . . . . . 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 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 4168 . . . . . . 7 𝐶 ⊆ (𝐵𝐶)
28 ssexg 5316 . . . . . . 7 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
2927, 16, 28sylancr 586 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐶 ∈ V)
30 inss2 4224 . . . . . 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 413 . . . . 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 1930 1 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  cun 3941  cin 3942  wss 3943   class class class wbr 5141   × cxp 5667  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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7974  df-2nd 7975  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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