Theorem unwdomg 9343

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 9342	. . 3 ⊢ (𝐴 ≼* 𝐵 → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
2	1	3ad2ant1 1132	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
3		brwdom3i 9342	. . . . 5 ⊢ (𝐶 ≼* 𝐷 → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
4	3	3ad2ant2 1133	. . . 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 9325	. . . . . . . . . 10 ⊢ Rel ≼*
7	6	brrelex1i 5643	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
8	6	brrelex1i 5643	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
9		unexg 7599	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
10	7, 8, 9	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 ∪ 𝐶) ∈ V)
11	10	3adant3 1131	. . . . . . 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 5644	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
14	6	brrelex2i 5644	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
15		unexg 7599	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
16	13, 14, 15	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
17	16	3adant3 1131	. . . . . . 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 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑏)))
21	20	rexbidv 3226	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏)))
22	21	rspcva 3559	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏))
23		fveq2 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑓‘𝑏) = (𝑓‘𝑧))
24	23	eqeq2d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑧)))
25	24	cbvrexvw 3384	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏) ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧))
26		ssun1 4106	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
27		iftrue 4465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑓)
28	27	fveq1d 6776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓‘𝑧))
29	28	eqeq2d 2749	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓‘𝑧)))
30	29	biimprd 247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = (𝑓‘𝑧) → 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
31	30	reximia 3176	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧) → ∃𝑧 ∈ 𝐵 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
32		ssrexv 3988	. . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . 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 712	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑏)))
40	39	rexbidv 3226	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏)))
41		fveq2 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑔‘𝑏) = (𝑔‘𝑧))
42	41	eqeq2d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑧)))
43	42	cbvrexvw 3384	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
44	40, 43	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)))
45	44	rspccva 3560	. . . . . . . . . . . . . 14 ⊢ ((∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
46		ssun2 4107	. . . . . . . . . . . . . . 15 ⊢ 𝐷 ⊆ (𝐵 ∪ 𝐷)
47		minel 4399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ¬ 𝑧 ∈ 𝐵)
48	47	ancoms 459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → ¬ 𝑧 ∈ 𝐵)
49	48	iffalsed 4470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑔)
50	49	fveq1d 6776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔‘𝑧))
51	50	eqeq2d 2749	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔‘𝑧)))
52	51	biimprd 247	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (𝑦 = (𝑔‘𝑧) → 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧)))
53	52	reximdva 3203	. . . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . 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 717	. . . . . . . . . . 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 955	. . . . . . . . . 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 594	. . . . . . . . 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 1134	. . . . . . 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 9339	. . . . 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 1936	. . 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 1938	1 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ifcif 4459   class class class wbr 5074  ‘cfv 6433   ≼^* cwdom 9323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-en 8734  df-dom 8735  df-sdom 8736  df-wdom 9324

This theorem is referenced by: (None)