Theorem unwdomg 9520

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 9519	. . 3 ⊢ (𝐴 ≼* 𝐵 → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ∃𝑓∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏))
3		brwdom3i 9519	. . . . 5 ⊢ (𝐶 ≼* 𝐷 → ∃𝑔∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏))
4	3	3ad2ant2 1134	. . . 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 9502	. . . . . . . . . 10 ⊢ Rel ≼*
7	6	brrelex1i 5688	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
8	6	brrelex1i 5688	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐶 ∈ V)
9		unexg 7683	. . . . . . . . 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 1132	. . . . . . 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 5689	. . . . . . . . 9 ⊢ (𝐴 ≼* 𝐵 → 𝐵 ∈ V)
14	6	brrelex2i 5689	. . . . . . . . 9 ⊢ (𝐶 ≼* 𝐷 → 𝐷 ∈ V)
15		unexg 7683	. . . . . . . . 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 1132	. . . . . . 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 4108	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐶) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶))
20		eqeq1 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑏)))
21	20	rexbidv 3175	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏)))
22	21	rspcva 3579	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 = (𝑓‘𝑏)) → ∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏))
23		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑓‘𝑏) = (𝑓‘𝑧))
24	23	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑓‘𝑏) ↔ 𝑦 = (𝑓‘𝑧)))
25	24	cbvrexvw 3226	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐵 𝑦 = (𝑓‘𝑏) ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧))
26		ssun1 4132	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
27		iftrue 4492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑓)
28	27	fveq1d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓‘𝑧))
29	28	eqeq2d 2747	. . . . . . . . . . . . . . . . . . 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 3084	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ 𝐵 𝑦 = (𝑓‘𝑧) → ∃𝑧 ∈ 𝐵 𝑦 = (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧))
32		ssrexv 4011	. . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . 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 712	. . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑏)))
40	39	rexbidv 3175	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏)))
41		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑧 → (𝑔‘𝑏) = (𝑔‘𝑧))
42	41	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑧 → (𝑦 = (𝑔‘𝑏) ↔ 𝑦 = (𝑔‘𝑧)))
43	42	cbvrexvw 3226	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ 𝐷 𝑦 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
44	40, 43	bitrdi 286	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ↔ ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧)))
45	44	rspccva 3580	. . . . . . . . . . . . . 14 ⊢ ((∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐷 𝑎 = (𝑔‘𝑏) ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐷 𝑦 = (𝑔‘𝑧))
46		ssun2 4133	. . . . . . . . . . . . . . 15 ⊢ 𝐷 ⊆ (𝐵 ∪ 𝐷)
47		minel 4425	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → ¬ 𝑧 ∈ 𝐵)
48	47	ancoms 459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → ¬ 𝑧 ∈ 𝐵)
49	48	iffalsed 4497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝐵, 𝑓, 𝑔) = 𝑔)
50	49	fveq1d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∩ 𝐷) = ∅ ∧ 𝑧 ∈ 𝐷) → (if(𝑧 ∈ 𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔‘𝑧))
51	50	eqeq2d 2747	. . . . . . . . . . . . . . . . . 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 3165	. . . . . . . . . . . . . . . 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 4011	. . . . . . . . . . . . . . 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 718	. . . . . . . . . . 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 956	. . . . . . . . . 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 1135	. . . . . . 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 9516	. . . . 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 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  ifcif 4486   class class class wbr 5105  ‘cfv 6496   ≼^* cwdom 9500

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-en 8884  df-dom 8885  df-sdom 8886  df-wdom 9501

This theorem is referenced by: (None)