Theorem brdom7disj 10530

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 10529	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑔(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
3		incom 4162	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
4		brdom7disj.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝐵) = ∅
5	3, 4	eqtri 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ 𝐴) = ∅
6		disjne 4415	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
7	5, 6	mp3an1 1477	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → 𝑥 ≠ 𝑤)
8		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
9		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
10		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
11		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
12	8, 9, 10, 11	opthpr 4818	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 𝑤 → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
13	7, 12	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ({𝑥, 𝑦} = {𝑧, 𝑤} ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
14		equcom 2051	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
15		equcom 2051	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
16	14, 15	anbi12i 640	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
17	13, 16	bitr2di 291	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
18		df-br 5112	. . . . . . . . . . . . . 14 ⊢ (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔)
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → (𝑧𝑔𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑔))
20	17, 19	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐴) → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
21	20	rexbidva 3189	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ ∃𝑤 ∈ 𝐴 ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
22	21	rexbidv 3191	. . . . . . . . . 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 3296	. . . . . . . . . . 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 5407	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
25		eqeq1 2769	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {𝑥, 𝑦} → (𝑣 = {𝑧, 𝑤} ↔ {𝑥, 𝑦} = {𝑧, 𝑤}))
26	25	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑥, 𝑦} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑥, 𝑦} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
27	26	2rexbidv 3232	. . . . . . . . . . . 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 3640	. . . . . . . . . . 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 281	. . . . . . . . . 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	bitr2di 291	. . . . . . . . 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 486	. . . . . . . 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 5114	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝑔𝑤 ↔ 𝑥𝑔𝑤))
33		breq2 5115	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥𝑔𝑤 ↔ 𝑥𝑔𝑦))
34	32, 33	ceqsrex2v 3619	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑥𝑔𝑦))
35	31, 34	bitrd 282	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ({𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑥𝑔𝑦))
36	35	rmobidva 3384	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦))
37	36	ralbiia 3111	. . . . 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 5407	. . . . . . . . . . 11 ⊢ {𝑦, 𝑥} ∈ V
39		eqeq1 2769	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑦, 𝑥} → (𝑣 = {𝑧, 𝑤} ↔ {𝑦, 𝑥} = {𝑧, 𝑤}))
40	39	anbi1d 643	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑦, 𝑥} → ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)))
41	40	2rexbidv 3232	. . . . . . . . . . 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 3640	. . . . . . . . . 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 4415	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∩ 𝐴) = ∅ ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
44	5, 43	mp3an1 1477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑧 ≠ 𝑥)
45	44	ancoms 464	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝑧 ≠ 𝑥)
46	10, 11, 9, 8	opthpr 4818	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ≠ 𝑥 → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
47	45, 46	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑧, 𝑤} = {𝑦, 𝑥} ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥)))
48		eqcom 2772	. . . . . . . . . . . . . 14 ⊢ ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ {𝑧, 𝑤} = {𝑦, 𝑥})
49		ancom 466	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
50	47, 48, 49	3bitr4g 317	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ({𝑦, 𝑥} = {𝑧, 𝑤} ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)))
51	18	bicomi 227	. . . . . . . . . . . . . 14 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤)
52	51	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 ↔ 𝑧𝑔𝑤))
53	50, 52	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
54	53	rexbidva 3189	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (∃𝑧 ∈ 𝐵 ({𝑦, 𝑥} = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) ↔ ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤)))
55	54	rexbidv 3191	. . . . . . . . . 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	bitrid 286	. . . . . . . . 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 486	. . . . . . . 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 5115	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑧𝑔𝑤 ↔ 𝑧𝑔𝑥))
59		breq1 5114	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧𝑔𝑥 ↔ 𝑦𝑔𝑥))
60	58, 59	ceqsrex2v 3619	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ∧ 𝑧𝑔𝑤) ↔ 𝑦𝑔𝑥))
61	57, 60	bitrd 282	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ 𝑦𝑔𝑥))
62	61	rexbidva 3189	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ↔ ∃𝑦 ∈ 𝐵 𝑦𝑔𝑥))
63	62	ralbiia 3111	. . . . 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 5412	. . . . . . . 8 ⊢ {{𝑧, 𝑤}} ∈ V
66		simpl 488	. . . . . . . . . 10 ⊢ ((𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔) → 𝑣 = {𝑧, 𝑤})
67	66	ss2abi 4021	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
68		df-sn 4592	. . . . . . . . 9 ⊢ {{𝑧, 𝑤}} = {𝑣 ∣ 𝑣 = {𝑧, 𝑤}}
69	67, 68	sseqtrri 3987	. . . . . . . 8 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ⊆ {{𝑧, 𝑤}}
70	65, 69	ssexi 5295	. . . . . . 7 ⊢ {𝑣 ∣ (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
71	64, 1, 70	ab2rexex2 7983	. . . . . 6 ⊢ {𝑣 ∣ ∃𝑤 ∈ 𝐴 ∃𝑧 ∈ 𝐵 (𝑣 = {𝑧, 𝑤} ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑔)} ∈ V
72		eleq2 2854	. . . . . . . . 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 3386	. . . . . . . 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 3190	. . . . . . 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 2854	. . . . . . . . 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 3191	. . . . . . . 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 3190	. . . . . . 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 644	. . . . . 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 3567	. . . . 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 611	. . . 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 1963	. . 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 4701	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤, 𝑧} = {𝑥, 𝑧})
83	82	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑧} ∈ 𝑓))
84		preq2 4702	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → {𝑥, 𝑧} = {𝑥, 𝑦})
85	84	eleq1d 2850	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ({𝑥, 𝑧} ∈ 𝑓 ↔ {𝑥, 𝑦} ∈ 𝑓))
86		eqid 2765	. . . . . . . 8 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}
87	8, 9, 83, 85, 86	brab 5530	. . . . . . 7 ⊢ (𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ {𝑥, 𝑦} ∈ 𝑓)
88	87	rmobii 3379	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓)
89	88	ralbii 3113	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦 ↔ ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓)
90		preq1 4701	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → {𝑤, 𝑧} = {𝑦, 𝑧})
91	90	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ({𝑤, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑧} ∈ 𝑓))
92		preq2 4702	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → {𝑦, 𝑧} = {𝑦, 𝑥})
93	92	eleq1d 2850	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ({𝑦, 𝑧} ∈ 𝑓 ↔ {𝑦, 𝑥} ∈ 𝑓))
94	9, 8, 91, 93, 86	brab 5530	. . . . . . 7 ⊢ (𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ {𝑦, 𝑥} ∈ 𝑓)
95	94	rexbii 3114	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
96	95	ralbii 3113	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)
97		df-opab 5176	. . . . . . 7 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)}
98		vuniex 7747	. . . . . . . 8 ⊢ ∪ 𝑓 ∈ V
99	11	prid1 4730	. . . . . . . . . . 11 ⊢ 𝑤 ∈ {𝑤, 𝑧}
100		elunii 4879	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
101	99, 100	mpan 703	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑤 ∈ ∪ 𝑓)
102	101	adantl 487	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
103	102	exlimiv 1963	. . . . . . . 8 ⊢ (∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑤 ∈ ∪ 𝑓)
104	10	prid2 4731	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑤, 𝑧}
105		elunii 4879	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑤, 𝑧} ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
106	104, 105	mpan 703	. . . . . . . . . 10 ⊢ ({𝑤, 𝑧} ∈ 𝑓 → 𝑧 ∈ ∪ 𝑓)
107	106	adantl 487	. . . . . . . . 9 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑧 ∈ ∪ 𝑓)
108		df-sn 4592	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} = {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
109		snex 5412	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩} ∈ V
110	108, 109	eqeltrri 2862	. . . . . . . . . 10 ⊢ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩} ∈ V
111		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓) → 𝑣 = ⟨𝑤, 𝑧⟩)
112	111	ss2abi 4021	. . . . . . . . . 10 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ⊆ {𝑣 ∣ 𝑣 = ⟨𝑤, 𝑧⟩}
113	110, 112	ssexi 5295	. . . . . . . . 9 ⊢ {𝑣 ∣ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
114	98, 107, 113	abexex 7974	. . . . . . . 8 ⊢ {𝑣 ∣ ∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
115	98, 103, 114	abexex 7974	. . . . . . 7 ⊢ {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ {𝑤, 𝑧} ∈ 𝑓)} ∈ V
116	97, 115	eqeltri 2861	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} ∈ V
117		breq 5113	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑥𝑔𝑦 ↔ 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
118	117	rmobidv 3386	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃*𝑦 ∈ 𝐴 𝑥𝑔𝑦 ↔ ∃*𝑦 ∈ 𝐴 𝑥{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑦))
119	118	ralbidv 3190	. . . . . . 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 5113	. . . . . . . . 9 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (𝑦𝑔𝑥 ↔ 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
121	120	rexbidv 3191	. . . . . . . 8 ⊢ (𝑔 = {⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓} → (∃𝑦 ∈ 𝐵 𝑦𝑔𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑦{⟨𝑤, 𝑧⟩ ∣ {𝑤, 𝑧} ∈ 𝑓}𝑥))
122	121	ralbidv 3190	. . . . . . 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 644	. . . . . 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 3567	. . . . 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 611	. . . 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 1963	. . 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 212	. 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 278	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  ∃*wrmo 3370  Vcvv 3457   ∩ cin 3905  ∅c0 4286  {csn 4591  {cpr 4593  ⟨cop 4597  ∪ cuni 4874   class class class wbr 5111  {copab 5175   ≼ cdom 8947

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-ac2 10462

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-card 9941  df-acn 9944  df-ac 10116

This theorem is used by: (None)