Theorem unwdomg 9559

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 9558	. . 3 ⊢ (𝐴 ≼* 𝐵 → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
2	1	3ad2ant1 1151	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
3		brwdom3i 9558	. . . . 5 ⊢ (𝐶 ≼* 𝐷 → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
4	3	3ad2ant2 1152	. . . 4 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
5	4	adantr 486	. . 3 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
6		relwdom 9541	. . . . . . . . . 10 ⊢ Rel ≼*
7	6	brrelex1i 5715	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
8	6	brrelex1i 5715	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
9		unexg 7748	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
10	7, 8, 9	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 ∪ 𝐶) ∈ V)
11	10	3adant3 1150	. . . . . . 7 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ∈ V)
12	11	adantr 486	. . . . . 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 5716	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
14	6	brrelex2i 5716	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
15		unexg 7748	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
16	13, 14, 15	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
17	16	3adant3 1150	. . . . . . 7 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
18	17	adantr 486	. . . . . 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 4103	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐶) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶))
20		eqeq1 2766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑏)))
21	20	rexbidv 3188	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏)))
22	21	rspcva 3577	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏))
23		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑓‘𝑏) = (𝑓‘𝑧))
24	23	eqeq2d 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑧)))
25	24	cbvrexvw 3243	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏) ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧))
26		ssun1 4127	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
27		iftrue 4491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑓)
28	27	fveq1d 6884	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓‘𝑧))
29	28	eqeq2d 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓‘𝑧)))
30	29	biimprd 251	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (𝑓‘𝑧) → 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
31	30	reximia 3099	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧) → ∃𝑧 ∈ 𝐵 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
32		ssrexv 4004	. . . . . . . . . . . . . . . . 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 69	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
34	25, 33	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
35	22, 34	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
36	35	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
37	36	adantlr 728	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑏)))
40	39	rexbidv 3188	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏)))
41		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑔‘𝑏) = (𝑔‘𝑧))
42	41	eqeq2d 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑧)))
43	42	cbvrexvw 3243	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
44	40, 43	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)))
45	44	rspccva 3578	. . . . . . . . . . . . . 14 ⊢ ((∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
46		ssun2 4128	. . . . . . . . . . . . . . 15 ⊢ 𝐷 ⊆ (𝐵 ∪ 𝐷)
47		minel 4422	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ¬ 𝑧 ∈ 𝐵)
48	47	ancoms 464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → ¬ 𝑧 ∈ 𝐵)
49	48	iffalsed 4496	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑔)
50	49	fveq1d 6884	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔‘𝑧))
51	50	eqeq2d 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔‘𝑧)))
52	51	biimprd 251	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (𝑦 = (𝑔‘𝑧) → 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
53	52	reximdva 3177	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∩ 𝐷) = ∅ → (∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧) → ∃𝑧 ∈ 𝐷 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
54	53	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)) → ∃𝑧 ∈ 𝐷 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
55		ssrexv 4004	. . . . . . . . . . . . . . 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 69	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)) → ∃𝑧 ∈ (𝐵 ∪ 𝐷)𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
57	45, 56	sylan2 605	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 733	. . . . . . . . . . 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 972	. . . . . . . . . 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 606	. . . . . . . . 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 463	. . . . . . . 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 1153	. . . . . . 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 461	. . . . . 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 9555	. . . . 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 462	. . . 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 1966	. . 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 16	. 2 ⊢ (((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))
69	2, 68	exlimddv 1968	1 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ifcif 4485   class class class wbr 5107  ‘cfv 6537   ≼^* cwdom 9539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-en 8956  df-dom 8957  df-sdom 8958  df-wdom 9540

This theorem is used by: (None)