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Theorem unxpwdom 9205
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 8632 . . . . 5 Rel ≼
21brrelex2i 5606 . . . 4 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐵𝐶) ∈ V)
3 domeng 8642 . . . 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 270 . 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 4188 . . . . . 6 (𝑥 ∩ (𝐵𝐶)) = ((𝑥𝐵) ∪ (𝑥𝐶))
8 simprr 773 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝑥 ⊆ (𝐵𝐶))
9 df-ss 3883 . . . . . . 7 (𝑥 ⊆ (𝐵𝐶) ↔ (𝑥 ∩ (𝐵𝐶)) = 𝑥)
108, 9sylib 221 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥 ∩ (𝐵𝐶)) = 𝑥)
117, 10eqtr3id 2792 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → ((𝑥𝐵) ∪ (𝑥𝐶)) = 𝑥)
126, 11breqtrrd 5081 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥𝐵) ∪ (𝑥𝐶)))
13 unxpwdom2 9204 . . . 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 4086 . . . . . . . 8 𝐵 ⊆ (𝐵𝐶)
162adantr 484 . . . . . . . 8 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐵𝐶) ∈ V)
17 ssexg 5216 . . . . . . . 8 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
1815, 16, 17sylancr 590 . . . . . . 7 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐵 ∈ V)
19 inss2 4144 . . . . . . 7 (𝑥𝐵) ⊆ 𝐵
20 ssdomg 8674 . . . . . . 7 (𝐵 ∈ V → ((𝑥𝐵) ⊆ 𝐵 → (𝑥𝐵) ≼ 𝐵))
2118, 19, 20mpisyl 21 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼ 𝐵)
22 domwdom 9190 . . . . . 6 ((𝑥𝐵) ≼ 𝐵 → (𝑥𝐵) ≼* 𝐵)
2321, 22syl 17 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐵) ≼* 𝐵)
24 wdomtr 9191 . . . . . 6 ((𝐴* (𝑥𝐵) ∧ (𝑥𝐵) ≼* 𝐵) → 𝐴* 𝐵)
2524expcom 417 . . . . 5 ((𝑥𝐵) ≼* 𝐵 → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
2623, 25syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴* (𝑥𝐵) → 𝐴* 𝐵))
27 ssun2 4087 . . . . . . 7 𝐶 ⊆ (𝐵𝐶)
28 ssexg 5216 . . . . . . 7 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
2927, 16, 28sylancr 590 . . . . . 6 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → 𝐶 ∈ V)
30 inss2 4144 . . . . . 6 (𝑥𝐶) ⊆ 𝐶
31 ssdomg 8674 . . . . . 6 (𝐶 ∈ V → ((𝑥𝐶) ⊆ 𝐶 → (𝑥𝐶) ≼ 𝐶))
3229, 30, 31mpisyl 21 . . . . 5 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝑥𝐶) ≼ 𝐶)
33 domtr 8681 . . . . . 6 ((𝐴 ≼ (𝑥𝐶) ∧ (𝑥𝐶) ≼ 𝐶) → 𝐴𝐶)
3433expcom 417 . . . . 5 ((𝑥𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3532, 34syl 17 . . . 4 (((𝐴 × 𝐴) ≼ (𝐵𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥𝑥 ⊆ (𝐵𝐶))) → (𝐴 ≼ (𝑥𝐶) → 𝐴𝐶))
3626, 35orim12d 965 . . 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 1943 1 ((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wex 1787  wcel 2110  Vcvv 3408  cun 3864  cin 3865  wss 3866   class class class wbr 5053   × cxp 5549  cen 8623  cdom 8624  * cwdom 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1st 7761  df-2nd 7762  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-wdom 9181
This theorem is referenced by:  pwdjudom  9830
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