Theorem brdom6disj 10297

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 10294	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
3		zfpair2 5354	. . . . . . . . 9 ⊢ {𝑥, 𝑦} ∈ V
4		eqeq1 2743	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
5	4	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
6		df-br 5076	. . . . . . . . . . . 12 ⊢ (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
7	6	anbi2i 623	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
8	5, 7	bitr4di 289	. . . . . . . . . 10 ⊢ (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤)))
9	8	2rexbidv 3230	. . . . . . . . 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 3610	. . . . . . . 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 4136	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
12		brdom7disj.3	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∩ 𝐵) = ∅
13	11, 12	eqtri 2767	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ 𝐴) = ∅
14		disjne 4389	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
15	13, 14	mp3an1 1447	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
16		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
17		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
18		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
19		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
20	16, 17, 18, 19	opthpr 4783	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≠ 𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
21	15, 20	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
22		breq12 5080	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥𝑔𝑦 ↔ 𝑧𝑔𝑤))
23	22	biimprd 247	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑧𝑔𝑤 → 𝑥𝑔𝑦))
24	21, 23	syl6bi 252	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} → (𝑧𝑔𝑤 → 𝑥𝑔𝑦)))
25	24	impd 411	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
26	25	ex 413	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑤 ∈ 𝐴 → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
27	26	adantrd 492	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦)))
28	27	rexlimdvv 3223	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ 𝑧𝑔𝑤) → 𝑥𝑔𝑦))
29	10, 28	syl5bi 241	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → 𝑥𝑔𝑦))
30	29	moimdv 2547	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (∃*𝑦 𝑥𝑔𝑦 → ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
31	30	ralimia 3086	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 → ∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
32		zfpair2 5354	. . . . . . . . . . . 12 ⊢ {𝑦, 𝑥} ∈ V
33		eqeq1 2743	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
34	33	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
35	34	2rexbidv 3230	. . . . . . . . . . . 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 3610	. . . . . . . . . . 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 4389	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
38	13, 37	mp3an1 1447	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
39	38	ancoms 459	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝑧 ≠ 𝑥)
40	18, 19, 17, 16	opthpr 4783	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≠ 𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
41	39, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
42		eqcom 2746	. . . . . . . . . . . . . . 15 ⊢ ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
43		ancom 461	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
44	41, 42, 43	3bitr4g 314	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)))
45	6	bicomi 223	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤)
46	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤))
47	44, 46	anbi12d 631	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
48	47	rexbidva 3226	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
49	48	rexbidv 3227	. . . . . . . . . . 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 282	. . . . . . . . . 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 481	. . . . . . . . 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 5079	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑧𝑔𝑤 ↔ 𝑧𝑔𝑥))
53		breq1 5078	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧𝑔𝑥 ↔ 𝑦𝑔𝑥))
54	52, 53	ceqsrex2v 3588	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
55	51, 54	bitrd 278	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
56	55	rexbidva 3226	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
57	56	ralbiia 3092	. . . . . 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 227	. . . . 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 5355	. . . . . . . 8 ⊢ {{𝑧, 𝑤}} ∈ V
61		simpl 483	. . . . . . . . . 10 ⊢ ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
62	61	ss2abi 4001	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
63		df-sn 4563	. . . . . . . . 9 ⊢ {{𝑧, 𝑤}} = {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
64	62, 63	sseqtrri 3959	. . . . . . . 8 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
65	60, 64	ssexi 5247	. . . . . . 7 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
66	59, 1, 65	ab2rexex2 7832	. . . . . 6 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
67		eleq2 2828	. . . . . . . . 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 2550	. . . . . . . 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 3113	. . . . . . 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 2828	. . . . . . . . 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 3227	. . . . . . . 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 3113	. . . . . . 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 631	. . . . . 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 3546	. . . . 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 596	. . . 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 1934	. . 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 4670	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
78	77	eleq1d 2824	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
79		preq2 4671	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
80	79	eleq1d 2824	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
81		eqid 2739	. . . . . . . 8 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
82	16, 17, 78, 80, 81	brab 5457	. . . . . . 7 ⊢ (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
83	82	mobii 2549	. . . . . 6 ⊢ (∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
84	83	ralbii 3093	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓)
85		preq1 4670	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
86	85	eleq1d 2824	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
87		preq2 4671	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
88	87	eleq1d 2824	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
89	17, 16, 86, 88, 81	brab 5457	. . . . . . 7 ⊢ (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
90	89	rexbii 3182	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
91	90	ralbii 3093	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
92		df-opab 5138	. . . . . . 7 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
93		vuniex 7601	. . . . . . . 8 ⊢ ∪ 𝑓 ∈ V
94	19	prid1 4699	. . . . . . . . . . 11 ⊢ 𝑤 ∈ {𝑤, 𝑧}
95		elunii 4845	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
96	94, 95	mpan 687	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑤 ∈ ∪ 𝑓)
97	96	adantl 482	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
98	97	exlimiv 1934	. . . . . . . 8 ⊢ (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
99	18	prid2 4700	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑤, 𝑧}
100		elunii 4845	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
101	99, 100	mpan 687	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑧 ∈ ∪ 𝑓)
102	101	adantl 482	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
103		df-sn 4563	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} = {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
104		snex 5355	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} ∈ V
105	103, 104	eqeltrri 2837	. . . . . . . . . 10 ⊢ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
106		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
107	106	ss2abi 4001	. . . . . . . . . 10 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
108	105, 107	ssexi 5247	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
109	93, 102, 108	abexex 7823	. . . . . . . 8 ⊢ {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
110	93, 98, 109	abexex 7823	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
111	92, 110	eqeltri 2836	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
112		breq 5077	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦 ↔ 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
113	112	mobidv 2550	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦 𝑥𝑔𝑦 ↔ ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
114	113	ralbidv 3113	. . . . . . 7 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑔𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
115		breq 5077	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥 ↔ 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
116	115	rexbidv 3227	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
117	116	ralbidv 3113	. . . . . . 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 631	. . . . . 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 3546	. . . . 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 599	. . . 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 1934	. . 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 208	. 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 274	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2107  ∃*wmo 2539  {cab 2716   ≠ wne 2944  ∀wral 3065  ∃wrex 3066  Vcvv 3433   ∩ cin 3887  ∅c0 4257  {csn 4562  {cpr 4564  ⟨cop 4568  ∪ cuni 4840   class class class wbr 5075  {copab 5137   ≼ cdom 8740

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-ac2 10228

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-er 8507  df-map 8626  df-en 8743  df-dom 8744  df-sdom 8745  df-card 9706  df-acn 9709  df-ac 9881

This theorem is referenced by:  grothprim  10599