Theorem unwdomg 9489

Description: Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
unwdomg	⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))

Proof of Theorem unwdomg

Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brwdom3i 9488	. . 3 ⊢ (𝐴 ≼* 𝐵 → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
2	1	3ad2ant1 1139	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
3		brwdom3i 9488	. . . . 5 ⊢ (𝐶 ≼* 𝐷 → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
4	3	3ad2ant2 1140	. . . 4 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
5	4	adantr 481	. . 3 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
6		relwdom 9471	. . . . . . . . . 10 ⊢ Rel ≼*
7	6	brrelex1i 5674	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
8	6	brrelex1i 5674	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
9		unexg 7686	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
10	7, 8, 9	syl2an 602	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 ∪ 𝐶) ∈ V)
11	10	3adant3 1138	. . . . . . 7 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ∈ V)
12	11	adantr 481	. . . . . 6 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) → (𝐴 ∪ 𝐶) ∈ V)
13	6	brrelex2i 5675	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
14	6	brrelex2i 5675	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
15		unexg 7686	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
16	13, 14, 15	syl2an 602	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
17	16	3adant3 1138	. . . . . . 7 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
18	17	adantr 481	. . . . . 6 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) → (𝐵 ∪ 𝐷) ∈ V)
19		elun 4083	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐶) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶))
20		eqeq1 2743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑏)))
21	20	rexbidv 3163	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏)))
22	21	rspcva 3558	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏))
23		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑓‘𝑏) = (𝑓‘𝑧))
24	23	eqeq2d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑧)))
25	24	cbvrexvw 3218	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏) ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧))
26		ssun1 4107	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
27		iftrue 4460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑓)
28	27	fveq1d 6829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓‘𝑧))
29	28	eqeq2d 2750	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓‘𝑧)))
30	29	biimprd 249	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (𝑓‘𝑧) → 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
31	30	reximia 3074	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧) → ∃𝑧 ∈ 𝐵 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
32		ssrexv 3984	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊆ (𝐵 ∪ 𝐷) → (∃𝑧 ∈ 𝐵 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
33	26, 31, 32	mpsyl 68	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
34	25, 33	sylbi 218	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
35	22, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
36	35	ancoms 459	. . . . . . . . . . . . 13 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
37	36	adantlr 721	. . . . . . . . . . . 12 ⊢ (((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏)) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
38	37	adantll 720	. . . . . . . . . . 11 ⊢ ((((𝐵 ∩ 𝐷) = ∅ ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
39		eqeq1 2743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑏)))
40	39	rexbidv 3163	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏)))
41		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑔‘𝑏) = (𝑔‘𝑧))
42	41	eqeq2d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑧)))
43	42	cbvrexvw 3218	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
44	40, 43	bitrdi 288	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)))
45	44	rspccva 3559	. . . . . . . . . . . . . 14 ⊢ ((∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
46		ssun2 4108	. . . . . . . . . . . . . . 15 ⊢ 𝐷 ⊆ (𝐵 ∪ 𝐷)
47		minel 4394	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ¬ 𝑧 ∈ 𝐵)
48	47	ancoms 459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → ¬ 𝑧 ∈ 𝐵)
49	48	iffalsed 4465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑔)
50	49	fveq1d 6829	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔‘𝑧))
51	50	eqeq2d 2750	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔‘𝑧)))
52	51	biimprd 249	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (𝑦 = (𝑔‘𝑧) → 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
53	52	reximdva 3152	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∩ 𝐷) = ∅ → (∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧) → ∃𝑧 ∈ 𝐷 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
54	53	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)) → ∃𝑧 ∈ 𝐷 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
55		ssrexv 3984	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ⊆ (𝐵 ∪ 𝐷) → (∃𝑧 ∈ 𝐷 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
56	46, 54, 55	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
57	45, 56	sylan2 599	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ (∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ∧ 𝑦 ∈ 𝐶)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
58	57	anassrs 468	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∩ 𝐷) = ∅ ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏)) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
59	58	adantlrl 726	. . . . . . . . . . 11 ⊢ ((((𝐵 ∩ 𝐷) = ∅ ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
60	38, 59	jaodan 965	. . . . . . . . . 10 ⊢ ((((𝐵 ∩ 𝐷) = ∅ ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
61	19, 60	sylan2b 600	. . . . . . . . 9 ⊢ ((((𝐵 ∩ 𝐷) = ∅ ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) ∧ 𝑦 ∈ (𝐴 ∪ 𝐶)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
62	61	expl 458	. . . . . . . 8 ⊢ ((𝐵 ∩ 𝐷) = ∅ → (((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏)) ∧ 𝑦 ∈ (𝐴 ∪ 𝐶)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
63	62	3ad2ant3 1141	. . . . . . 7 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏)) ∧ 𝑦 ∈ (𝐴 ∪ 𝐶)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
64	63	impl 456	. . . . . 6 ⊢ ((((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) ∧ 𝑦 ∈ (𝐴 ∪ 𝐶)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
65	12, 18, 64	wdom2d 9485	. . . . 5 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))
66	65	expr 457	. . . 4 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → (∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷)))
67	66	exlimdv 1940	. . 3 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → (∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷)))
68	5, 67	mpd 15	. 2 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))
69	2, 68	exlimddv 1942	1 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ifcif 4454   class class class wbr 5072  ‘cfv 6485   ≼^* cwdom 9469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-en 8884  df-dom 8885  df-sdom 8886  df-wdom 9470

This theorem is referenced by: (None)