Theorem exmidfodomrlemim 7194

Description: Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)

Assertion

Ref	Expression
exmidfodomrlemim	⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))

Distinct variable groups:   𝑥,𝑓,𝑧   𝑦,𝑓,𝑧

Proof of Theorem exmidfodomrlemim

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

Step	Hyp	Ref	Expression
1		brdomi 6743	. . . . 5 ⊢ (𝑦 ≼ 𝑥 → ∃𝑔 𝑔:𝑦–1-1→𝑥)
2	1	ad2antll 491	. . . 4 ⊢ ((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) → ∃𝑔 𝑔:𝑦–1-1→𝑥)
3		df-f1 5217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝑦–1-1→𝑥 ↔ (𝑔:𝑦⟶𝑥 ∧ Fun ◡𝑔))
4	3	simprbi 275	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → Fun ◡𝑔)
5		vex 2740	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
6	5	fconst 5407	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧}
7		ffun 5364	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝑥 ∖ ran 𝑔) × {𝑧}))
8	6, 7	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})
9	4, 8	jctir 313	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑦–1-1→𝑥 → (Fun ◡𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})))
10		df-rn 4634	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑔 = dom ◡𝑔
11	10	eqcomi 2181	. . . . . . . . . . . . . . . . 17 ⊢ dom ◡𝑔 = ran 𝑔
12	5	snm 3711	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑤 𝑤 ∈ {𝑧}
13		dmxpm 4843	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑤 𝑤 ∈ {𝑧} → dom ((𝑥 ∖ ran 𝑔) × {𝑧}) = (𝑥 ∖ ran 𝑔))
14	12, 13	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ dom ((𝑥 ∖ ran 𝑔) × {𝑧}) = (𝑥 ∖ ran 𝑔)
15	11, 14	ineq12i 3334	. . . . . . . . . . . . . . . 16 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔))
16		disjdif 3495	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔)) = ∅
17	15, 16	eqtri 2198	. . . . . . . . . . . . . . 15 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅
18		funun 5256	. . . . . . . . . . . . . . 15 ⊢ (((Fun ◡𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})) ∧ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
19	9, 17, 18	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑦–1-1→𝑥 → Fun (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
20	19	adantl 277	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → Fun (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
21		dmun 4830	. . . . . . . . . . . . . . 15 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
22	10	uneq1i 3285	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
23	14	uneq2i 3286	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
24	21, 22, 23	3eqtr2i 2204	. . . . . . . . . . . . . 14 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
25		f1rn 5418	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → ran 𝑔 ⊆ 𝑥)
26	25	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran 𝑔 ⊆ 𝑥)
27		exmidexmid 4193	. . . . . . . . . . . . . . . . 17 ⊢ (EXMID → DECID 𝑢 ∈ ran 𝑔)
28	27	ralrimivw 2551	. . . . . . . . . . . . . . . 16 ⊢ (EXMID → ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔)
29	28	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔)
30		undifdcss 6916	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)) ↔ (ran 𝑔 ⊆ 𝑥 ∧ ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔))
31	26, 29, 30	sylanbrc 417	. . . . . . . . . . . . . 14 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → 𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)))
32	24, 31	eqtr4id 2229	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥)
33		df-fn 5215	. . . . . . . . . . . . 13 ⊢ ((◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥 ↔ (Fun (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∧ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥))
34	20, 32, 33	sylanbrc 417	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥)
35		rnun 5033	. . . . . . . . . . . . 13 ⊢ ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧}))
36		dfdm4 4815	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑔 = ran ◡𝑔
37		f1dm 5422	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → dom 𝑔 = 𝑦)
38	36, 37	eqtr3id 2224	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑦–1-1→𝑥 → ran ◡𝑔 = 𝑦)
39	38	uneq1d 3288	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑦–1-1→𝑥 → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})))
40		exmidexmid 4193	. . . . . . . . . . . . . . . . . 18 ⊢ (EXMID → DECID ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔))
41		exmiddc 836	. . . . . . . . . . . . . . . . . 18 ⊢ (DECID ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∨ ¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔)))
42	40, 41	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (EXMID → (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∨ ¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔)))
43		rnxpm 5054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
44	43	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
45		snssi 3735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦)
46	45	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → {𝑧} ⊆ 𝑦)
47	44, 46	eqsstrd 3191	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
48	47	ex 115	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
49		notm0 3443	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ↔ (𝑥 ∖ ran 𝑔) = ∅)
50		xpeq1 4637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
51		0xp 4703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅ × {𝑧}) = ∅
52	50, 51	eqtrdi 2226	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
53	52	rneqd 4852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ran ∅)
54		rn0 4879	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ran ∅ = ∅
55	53, 54	eqtrdi 2226	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
56		0ss 3461	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ⊆ 𝑦
57	55, 56	eqsstrdi 3207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
58	57	a1d 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
59	49, 58	sylbi 121	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
60	48, 59	jaoi 716	. . . . . . . . . . . . . . . . 17 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∨ ¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔)) → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
61	42, 60	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (EXMID → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
62	61	imp 124	. . . . . . . . . . . . . . 15 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
63		ssequn2 3308	. . . . . . . . . . . . . . 15 ⊢ (ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦 ↔ (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
64	62, 63	sylib 122	. . . . . . . . . . . . . 14 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
65	39, 64	sylan9eqr 2232	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
66	35, 65	eqtrid 2222	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
67		df-fo 5218	. . . . . . . . . . . 12 ⊢ ((◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦 ↔ ((◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥 ∧ ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦))
68	34, 66, 67	sylanbrc 417	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦)
69		vex 2740	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
70	69	cnvex 5163	. . . . . . . . . . . . 13 ⊢ ◡𝑔 ∈ V
71		vex 2740	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
72		difexg 4141	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∖ ran 𝑔) ∈ V)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ ran 𝑔) ∈ V
74	5	snex 4182	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ V
75	73, 74	xpex 4738	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}) ∈ V
76	70, 75	unex 4438	. . . . . . . . . . . 12 ⊢ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∈ V
77		foeq1 5430	. . . . . . . . . . . 12 ⊢ (𝑓 = (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝑥–onto→𝑦 ↔ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦))
78	76, 77	spcev 2832	. . . . . . . . . . 11 ⊢ ((◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦 → ∃𝑓 𝑓:𝑥–onto→𝑦)
79	68, 78	syl 14	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
80	79	an32s 568	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑔:𝑦–1-1→𝑥) ∧ 𝑧 ∈ 𝑦) → ∃𝑓 𝑓:𝑥–onto→𝑦)
81	80	ex 115	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑔:𝑦–1-1→𝑥) → (𝑧 ∈ 𝑦 → ∃𝑓 𝑓:𝑥–onto→𝑦))
82	81	exlimdv 1819	. . . . . . 7 ⊢ ((EXMID ∧ 𝑔:𝑦–1-1→𝑥) → (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑓 𝑓:𝑥–onto→𝑦))
83	82	imp 124	. . . . . 6 ⊢ (((EXMID ∧ 𝑔:𝑦–1-1→𝑥) ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑓 𝑓:𝑥–onto→𝑦)
84	83	an32s 568	. . . . 5 ⊢ (((EXMID ∧ ∃𝑧 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
85	84	adantlrr 483	. . . 4 ⊢ (((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
86	2, 85	exlimddv 1898	. . 3 ⊢ ((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) → ∃𝑓 𝑓:𝑥–onto→𝑦)
87	86	ex 115	. 2 ⊢ (EXMID → ((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))
88	87	alrimivv 1875	1 ⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ∖ cdif 3126   ∪ cun 3127   ∩ cin 3128   ⊆ wss 3129  ∅c0 3422  {csn 3591   class class class wbr 4000  EXMIDwem 4191   × cxp 4621  ^◡ccnv 4622  dom cdm 4623  ran crn 4624  Fun wfun 5206   Fn wfn 5207  ⟶wf 5208  –1-1→wf1 5209  –onto→wfo 5210   ≼ cdom 6733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-exmid 4192  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-dom 6736

This theorem is referenced by:  exmidfodomr  7197