Theorem brdom6disj 10538

Description: An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Hypotheses

Ref	Expression
brdom7disj.1	⊢ 𝐴 ∈ V
brdom7disj.2	⊢ 𝐵 ∈ V
brdom7disj.3	⊢ (𝐴 ∩ 𝐵) = ∅

Assertion

Ref	Expression
brdom6disj	⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem brdom6disj

Dummy variables 𝑔 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdom7disj.2	. . 3 ⊢ 𝐵 ∈ V
2	1	brdom5 10535	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
3		zfpair2 5403	. . . . . . . . 9 ⊢ {𝑥, 𝑦} ∈ V
4		eqeq1 2766	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
5	4	anbi1d 643	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
6		df-br 5108	. . . . . . . . . . . 12 ⊢ (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
7	6	anbi2i 635	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
8	5, 7	bitr4di 292	. . . . . . . . . 10 ⊢ (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
9	8	2rexbidv 3229	. . . . . . . . 9 ⊢ (𝑣 = {𝑥, 𝑦} → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
10	3, 9	elab 3636	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤))
11		incom 4158	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
12		brdom7disj.3	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∩ 𝐵) = ∅
13	11, 12	eqtri 2785	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ 𝐴) = ∅
14		disjne 4411	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
15	13, 14	mp3an1 1477	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
16		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
17		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
18		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
19		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
20	16, 17, 18, 19	opthpr 4814	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≠ 𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
21	15, 20	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
22		breq12 5112	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝑔𝑦 ↔ 𝑧𝑔𝑤))
23	22	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑧𝑔𝑤 → 𝑥𝑔𝑦))
24	21, 23	biimtrdi 256	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} → (𝑧𝑔𝑤 → 𝑥𝑔𝑦)))
25	24	impd 416	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
26	25	ex 418	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑤 ∈ 𝐴 → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
27	26	adantrd 497	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
28	27	rexlimdvv 3220	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
29	10, 28	biimtrid 245	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → 𝑥𝑔𝑦))
30	29	moimdv 2573	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (∃*𝑦 𝑥𝑔𝑦 → ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
31	30	ralimia 3098	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 → ∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
32		zfpair2 5403	. . . . . . . . . . . 12 ⊢ {𝑦, 𝑥} ∈ V
33		eqeq1 2766	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
34	33	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
35	34	2rexbidv 3229	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑦, 𝑥} → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
36	32, 35	elab 3636	. . . . . . . . . . 11 ⊢ ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
37		disjne 4411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
38	13, 37	mp3an1 1477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
39	38	ancoms 464	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝑧 ≠ 𝑥)
40	18, 19, 17, 16	opthpr 4814	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≠ 𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
41	39, 40	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
42		eqcom 2769	. . . . . . . . . . . . . . 15 ⊢ ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
43		ancom 466	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
44	41, 42, 43	3bitr4g 317	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)))
45	6	bicomi 227	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤)
46	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤))
47	44, 46	anbi12d 644	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
48	47	rexbidva 3186	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
49	48	rexbidv 3188	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
50	36, 49	bitrid 286	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
51	50	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
52		breq2 5111	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑧𝑔𝑤 ↔ 𝑧𝑔𝑥))
53		breq1 5110	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧𝑔𝑥 ↔ 𝑦𝑔𝑥))
54	52, 53	ceqsrex2v 3615	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
55	51, 54	bitrd 282	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
56	55	rexbidva 3186	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
57	56	ralbiia 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥)
58	57	biimpri 231	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
59		brdom7disj.1	. . . . . . 7 ⊢ 𝐴 ∈ V
60		snex 5408	. . . . . . . 8 ⊢ {{𝑧, 𝑤}} ∈ V
61		simpl 488	. . . . . . . . . 10 ⊢ ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
62	61	ss2abi 4017	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
63		df-sn 4588	. . . . . . . . 9 ⊢ {{𝑧, 𝑤}} = {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
64	62, 63	sseqtrri 3983	. . . . . . . 8 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
65	60, 64	ssexi 5291	. . . . . . 7 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
66	59, 1, 65	ab2rexex2 7980	. . . . . 6 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
67		eleq2 2851	. . . . . . . . 9 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑥, 𝑦} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
68	67	mobidv 2576	. . . . . . . 8 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ↔ ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
69	68	ralbidv 3187	. . . . . . 7 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
70		eleq2 2851	. . . . . . . . 9 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ({𝑦, 𝑥} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
71	70	rexbidv 3188	. . . . . . . 8 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓 ↔ ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
72	71	ralbidv 3187	. . . . . . 7 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
73	69, 72	anbi12d 644	. . . . . 6 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ((∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓) ↔ (∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})))
74	66, 73	spcev 3563	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}) → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
75	31, 58, 74	syl2an 608	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
76	75	exlimiv 1963	. . 3 ⊢ (∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
77		preq1 4697	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
78	77	eleq1d 2847	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
79		preq2 4698	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
80	79	eleq1d 2847	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
81		eqid 2762	. . . . . . . 8 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
82	16, 17, 78, 80, 81	brab 5526	. . . . . . 7 ⊢ (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
83	82	mobii 2575	. . . . . 6 ⊢ (∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
84	83	ralbii 3110	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
85		preq1 4697	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
86	85	eleq1d 2847	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
87		preq2 4698	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
88	87	eleq1d 2847	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
89	17, 16, 86, 88, 81	brab 5526	. . . . . . 7 ⊢ (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
90	89	rexbii 3111	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
91	90	ralbii 3110	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
92		df-opab 5172	. . . . . . 7 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
93		vuniex 7744	. . . . . . . 8 ⊢ ∪ 𝑓 ∈ V
94	19	prid1 4726	. . . . . . . . . . 11 ⊢ 𝑤 ∈ {𝑤, 𝑧}
95		elunii 4875	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
96	94, 95	mpan 703	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑤 ∈ ∪ 𝑓)
97	96	adantl 487	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
98	97	exlimiv 1963	. . . . . . . 8 ⊢ (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
99	18	prid2 4727	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑤, 𝑧}
100		elunii 4875	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
101	99, 100	mpan 703	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑧 ∈ ∪ 𝑓)
102	101	adantl 487	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
103		df-sn 4588	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} = {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
104		snex 5408	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} ∈ V
105	103, 104	eqeltrri 2859	. . . . . . . . . 10 ⊢ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
106		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
107	106	ss2abi 4017	. . . . . . . . . 10 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
108	105, 107	ssexi 5291	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
109	93, 102, 108	abexex 7971	. . . . . . . 8 ⊢ {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
110	93, 98, 109	abexex 7971	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
111	92, 110	eqeltri 2858	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
112		breq 5109	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦 ↔ 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
113	112	mobidv 2576	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦 𝑥𝑔𝑦 ↔ ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
114	113	ralbidv 3187	. . . . . . 7 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
115		breq 5109	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥 ↔ 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
116	115	rexbidv 3188	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
117	116	ralbidv 3187	. . . . . . 7 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
118	114, 117	anbi12d 644	. . . . . 6 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → ((∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) ↔ (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥)))
119	111, 118	spcev 3563	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥) → ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
120	84, 91, 119	syl2anbr 611	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
121	120	exlimiv 1963	. . 3 ⊢ (∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
122	76, 121	impbii 212	. 2 ⊢ (∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
123	2, 122	bitri 278	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2564  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∩ cin 3901  ∅c0 4282  {csn 4587  {cpr 4589  ⟨cop 4593  ∪ cuni 4870   class class class wbr 5107  {copab 5171   ≼ cdom 8953

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-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-ac2 10468

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-rmo 3367  df-reu 3368  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-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  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-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  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-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-card 9947  df-acn 9950  df-ac 10122

This theorem is used by:  grothprim  10844