Theorem unxpwdom 9603

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 8965	. . . . 5 ⊢ Rel ≼
2	1	brrelex2i 5711	. . . 4 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
3		domeng 8977	. . . 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 770	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ 𝑥)
7		indi 4259	. . . . . 6 ⊢ (𝑥 ∩ (𝐵 ∪ 𝐶)) = ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶))
8		simprr 772	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝑥 ⊆ (𝐵 ∪ 𝐶))
9		dfss2 3944	. . . . . . 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 2784	. . . . 5 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)) = 𝑥)
12	6, 11	breqtrrd 5147	. . . 4 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐴 × 𝐴) ≈ ((𝑥 ∩ 𝐵) ∪ (𝑥 ∩ 𝐶)))
13		unxpwdom2 9602	. . . 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 4153	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
16	2	adantr 480	. . . . . . . 8 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → (𝐵 ∪ 𝐶) ∈ V)
17		ssexg 5293	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V)
18	15, 16, 17	sylancr 587	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐵 ∈ V)
19		inss2 4213	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
20		ssdomg 9014	. . . . . . 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 9588	. . . . . 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 9589	. . . . . 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 4154	. . . . . . 7 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
28		ssexg 5293	. . . . . . 7 ⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V)
29	27, 16, 28	sylancr 587	. . . . . 6 ⊢ (((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) ∧ ((𝐴 × 𝐴) ≈ 𝑥 ∧ 𝑥 ⊆ (𝐵 ∪ 𝐶))) → 𝐶 ∈ V)
30		inss2 4213	. . . . . 6 ⊢ (𝑥 ∩ 𝐶) ⊆ 𝐶
31		ssdomg 9014	. . . . . 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 9021	. . . . . 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 966	. . 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 1935	1 ⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926   class class class wbr 5119   × cxp 5652   ≈ cen 8956   ≼ cdom 8957   ≼^* cwdom 9578

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-1st 7988  df-2nd 7989  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-wdom 9579

This theorem is referenced by:  pwdjudom  10229