Theorem unxpwdom 9506

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) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Proof of Theorem unxpwdom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 8901	. . . . 5 ⊢ Rel ≼
2	1	brrelex2i 5689	. . . 4 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
3		domeng 8911	. . . 4 ⊢ ((𝐵 ∪ 𝐶) ∈ V → ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))))
4	2, 3	syl 17	. . 3 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ↔ ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))))
5	4	ibi 267	. 2 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → ∃𝑥((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶)))
6		simprl 771	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7		indi 4238	. . . . . 6 ⊢ (𝑥 ∩ (𝐵 ∪ 𝐶)) = ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶))
8		simprr 773	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝑥 ⊆ (𝐵 ∪ 𝐶))
9		dfss2 3921	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐵 ∪ 𝐶) ↔ (𝑥 ∩ (𝐵 ∪ 𝐶)) = 𝑥)
10	8, 9	sylib 218	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ (𝐵 ∪ 𝐶)) = 𝑥)
11	7, 10	eqtr3id 2786	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) = 𝑥)
12	6, 11	breqtrrd 5128	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)))
13		unxpwdom2 9505	. . . 4 ⊢ ((𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) → (𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)))
14	12, 13	syl 17	. . 3 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)))
15		ssun1 4132	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
16	2	adantr 480	. . . . . . . 8 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐵 ∪ 𝐶) ∈ V)
17		ssexg 5270	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V)
18	15, 16, 17	sylancr 588	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐵 ∈ V)
19		inss2 4192	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
20		ssdomg 8949	. . . . . . 7 ⊢ (𝐵 ∈ V → ((𝑥 ∩ 𝐵) ⊆ 𝐵 → (𝑥 ∩ 𝐵) ≼ 𝐵))
21	18, 19, 20	mpisyl 21	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐵) ≼ 𝐵)
22		domwdom 9491	. . . . . 6 ⊢ ((𝑥 ∩ 𝐵) ≼ 𝐵 → (𝑥 ∩ 𝐵) ≼* 𝐵)
23	21, 22	syl 17	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐵) ≼* 𝐵)
24		wdomtr 9492	. . . . . 6 ⊢ ((𝐴 ≼* (𝑥 ∩ 𝐵) ∧ (𝑥 ∩ 𝐵) ≼* 𝐵) → 𝐴 ≼* 𝐵)
25	24	expcom 413	. . . . 5 ⊢ ((𝑥 ∩ 𝐵) ≼* 𝐵 → (𝐴 ≼* (𝑥 ∩ 𝐵) → 𝐴 ≼* 𝐵))
26	23, 25	syl 17	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* (𝑥 ∩ 𝐵) → 𝐴 ≼* 𝐵))
27		ssun2 4133	. . . . . . 7 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
28		ssexg 5270	. . . . . . 7 ⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V)
29	27, 16, 28	sylancr 588	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐶 ∈ V)
30		inss2 4192	. . . . . 6 ⊢ (𝑥 ∩ 𝐶) ⊆ 𝐶
31		ssdomg 8949	. . . . . 6 ⊢ (𝐶 ∈ V → ((𝑥 ∩ 𝐶) ⊆ 𝐶 → (𝑥 ∩ 𝐶) ≼ 𝐶))
32	29, 30, 31	mpisyl 21	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝑥 ∩ 𝐶) ≼ 𝐶)
33		domtr 8956	. . . . . 6 ⊢ ((𝐴 ≼ (𝑥 ∩ 𝐶) ∧ (𝑥 ∩ 𝐶) ≼ 𝐶) → 𝐴 ≼ 𝐶)
34	33	expcom 413	. . . . 5 ⊢ ((𝑥 ∩ 𝐶) ≼ 𝐶 → (𝐴 ≼ (𝑥 ∩ 𝐶) → 𝐴 ≼ 𝐶))
35	32, 34	syl 17	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼ (𝑥 ∩ 𝐶) → 𝐴 ≼ 𝐶))
36	26, 35	orim12d 967	. . 3 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝐴 ≼* (𝑥 ∩ 𝐵) ∨ 𝐴 ≼ (𝑥 ∩ 𝐶)) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)))
37	14, 36	mpd 15	. 2 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))
38	5, 37	exlimddv 1937	1 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903   class class class wbr 5100   × cxp 5630   ≈ cen 8892   ≼ cdom 8893   ≼^* cwdom 9481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7943  df-2nd 7944  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-wdom 9482

This theorem is referenced by:  pwdjudom  10137