Theorem brdom7disj 9234

Description: An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Hypotheses

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

Assertion

Ref	Expression
brdom7disj	⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))

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

Proof of Theorem brdom7disj

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	brdom4 9233	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
3		incom 3767	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
4		brdom7disj.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝐵) = ∅
5	3, 4	eqtri 2632	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ 𝐴) = ∅
6		disjne 3974	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
7	5, 6	mp3an1 1403	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
8		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
9		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
11		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
12	8, 9, 10, 11	opthpr 4324	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
13	7, 12	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
14		equcom 1932	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
15		equcom 1932	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
16	14, 15	anbi12i 729	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
17	13, 16	syl6rbb 276	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
18		df-br 4584	. . . . . . . . . . . . . 14 ⊢ (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
20	17, 19	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
21	20	rexbidva 3031	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ∃𝑤 ∈ 𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
22	21	rexbidv 3034	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
23		rexcom 3080	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
24		zfpair2 4834	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
25		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
26	25	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
27	26	2rexbidv 3039	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑥, 𝑦} → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
28	24, 27	elab 3319	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
29	23, 28	bitr4i 266	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)})
30	22, 29	syl6rbb 276	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤)))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤)))
32		breq1 4586	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝑔𝑤 ↔ 𝑥𝑔𝑤))
33		breq2 4587	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥𝑔𝑤 ↔ 𝑥𝑔𝑦))
34	32, 33	ceqsrex2v 3308	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑥𝑔𝑦))
35	31, 34	bitrd 267	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑥𝑔𝑦))
36	35	rmobidva 3107	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦))
37	36	ralbiia 2962	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦)
38		zfpair2 4834	. . . . . . . . . . 11 ⊢ {𝑦, 𝑥} ∈ V
39		eqeq1 2614	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
40	39	anbi1d 737	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
41	40	2rexbidv 3039	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑦, 𝑥} → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
42	38, 41	elab 3319	. . . . . . . . . 10 ⊢ ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
43		disjne 3974	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
44	5, 43	mp3an1 1403	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
45	44	ancoms 468	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝑧 ≠ 𝑥)
46	10, 11, 9, 8	opthpr 4324	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ≠ 𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
47	45, 46	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
48		eqcom 2617	. . . . . . . . . . . . . 14 ⊢ ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
49		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
50	47, 48, 49	3bitr4g 302	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)))
51	18	bicomi 213	. . . . . . . . . . . . . 14 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤)
52	51	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤))
53	50, 52	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
54	53	rexbidva 3031	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
55	54	rexbidv 3034	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
56	42, 55	syl5bb 271	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
57	56	adantr 480	. . . . . . . 8 ⊢ ((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)))
58		breq2 4587	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑧𝑔𝑤 ↔ 𝑧𝑔𝑥))
59		breq1 4586	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧𝑔𝑥 ↔ 𝑦𝑔𝑥))
60	58, 59	ceqsrex2v 3308	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
61	57, 60	bitrd 267	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
62	61	rexbidva 3031	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
63	62	ralbiia 2962	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥)
64		brdom7disj.1	. . . . . . 7 ⊢ 𝐴 ∈ V
65		snex 4835	. . . . . . . 8 ⊢ {{𝑧, 𝑤}} ∈ V
66		simpl 472	. . . . . . . . . 10 ⊢ ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
67	66	ss2abi 3637	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
68		df-sn 4126	. . . . . . . . 9 ⊢ {{𝑧, 𝑤}} = {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
69	67, 68	sseqtr4i 3601	. . . . . . . 8 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
70	65, 69	ssexi 4731	. . . . . . 7 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
71	64, 1, 70	ab2rexex2 7051	. . . . . 6 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
72		eleq2 2677	. . . . . . . . 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)}))
73	72	rmobidv 3108	. . . . . . . 8 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ↔ ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
74	73	ralbidv 2969	. . . . . . 7 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)}))
75		eleq2 2677	. . . . . . . . 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)}))
76	75	rexbidv 3034	. . . . . . . 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)}))
77	76	ralbidv 2969	. . . . . . 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)}))
78	74, 77	anbi12d 743	. . . . . 6 ⊢ (𝑓 = {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} → ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓) ↔ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {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)})))
79	71, 78	spcev 3273	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {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 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
80	37, 63, 79	syl2anbr 496	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
81	80	exlimiv 1845	. . 3 ⊢ (∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
82		preq1 4212	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
83	82	eleq1d 2672	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
84		preq2 4213	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
85	84	eleq1d 2672	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
86		eqid 2610	. . . . . . . 8 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
87	8, 9, 83, 85, 86	brab 4923	. . . . . . 7 ⊢ (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
88	87	rmobii 3110	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓)
89	88	ralbii 2963	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓)
90		preq1 4212	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
91	90	eleq1d 2672	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
92		preq2 4213	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
93	92	eleq1d 2672	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
94	9, 8, 91, 93, 86	brab 4923	. . . . . . 7 ⊢ (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
95	94	rexbii 3023	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
96	95	ralbii 2963	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
97		df-opab 4644	. . . . . . 7 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
98		vuniex 6852	. . . . . . . 8 ⊢ ∪ 𝑓 ∈ V
99	11	prid1 4241	. . . . . . . . . . 11 ⊢ 𝑤 ∈ {𝑤, 𝑧}
100		elunii 4377	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
101	99, 100	mpan 702	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑤 ∈ ∪ 𝑓)
102	101	adantl 481	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
103	102	exlimiv 1845	. . . . . . . 8 ⊢ (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
104	10	prid2 4242	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑤, 𝑧}
105		elunii 4377	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
106	104, 105	mpan 702	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑧 ∈ ∪ 𝑓)
107	106	adantl 481	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
108		df-sn 4126	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} = {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
109		snex 4835	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} ∈ V
110	108, 109	eqeltrri 2685	. . . . . . . . . 10 ⊢ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
111		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
112	111	ss2abi 3637	. . . . . . . . . 10 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
113	110, 112	ssexi 4731	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
114	98, 107, 113	abexex 7042	. . . . . . . 8 ⊢ {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
115	98, 103, 114	abexex 7042	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
116	97, 115	eqeltri 2684	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
117		breq 4585	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦 ↔ 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
118	117	rmobidv 3108	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ↔ ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
119	118	ralbidv 2969	. . . . . . 7 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
120		breq 4585	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥 ↔ 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
121	120	rexbidv 3034	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
122	121	ralbidv 2969	. . . . . . 7 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
123	119, 122	anbi12d 743	. . . . . 6 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) ↔ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥)))
124	116, 123	spcev 3273	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥) → ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
125	89, 96, 124	syl2anbr 496	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
126	125	exlimiv 1845	. . 3 ⊢ (∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓) → ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
127	81, 126	impbii 198	. 2 ⊢ (∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥) ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
128	2, 127	bitri 263	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃*wrmo 2899  Vcvv 3173   ∩ cin 3539  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583  {copab 4642   ≼ cdom 7839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648  df-acn 8651  df-ac 8822

This theorem is referenced by: (None)