Theorem exmidfodomrlemim 7178

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 6727	. . . . 5 ⊢ (𝑦 ≼ 𝑥 → ∃𝑔 𝑔:𝑦–1-1→𝑥)
2	1	ad2antll 488	. . . 4 ⊢ ((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) → ∃𝑔 𝑔:𝑦–1-1→𝑥)
3		df-f1 5203	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝑦–1-1→𝑥 ↔ (𝑔:𝑦⟶𝑥 ∧ Fun ◡𝑔))
4	3	simprbi 273	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → Fun ◡𝑔)
5		vex 2733	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
6	5	fconst 5393	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧}
7		ffun 5350	. . . . . . . . . . . . . . . . 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 311	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑦–1-1→𝑥 → (Fun ◡𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})))
10		df-rn 4622	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑔 = dom ◡𝑔
11	10	eqcomi 2174	. . . . . . . . . . . . . . . . 17 ⊢ dom ◡𝑔 = ran 𝑔
12	5	snm 3703	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑤 𝑤 ∈ {𝑧}
13		dmxpm 4831	. . . . . . . . . . . . . . . . . 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 3326	. . . . . . . . . . . . . . . 16 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔))
16		disjdif 3487	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔)) = ∅
17	15, 16	eqtri 2191	. . . . . . . . . . . . . . 15 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅
18		funun 5242	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑦–1-1→𝑥 → Fun (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
20	19	adantl 275	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → Fun (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
21		dmun 4818	. . . . . . . . . . . . . . 15 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
22	10	uneq1i 3277	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
23	14	uneq2i 3278	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
24	21, 22, 23	3eqtr2i 2197	. . . . . . . . . . . . . 14 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
25		f1rn 5404	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → ran 𝑔 ⊆ 𝑥)
26	25	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran 𝑔 ⊆ 𝑥)
27		exmidexmid 4182	. . . . . . . . . . . . . . . . 17 ⊢ (EXMID → DECID 𝑢 ∈ ran 𝑔)
28	27	ralrimivw 2544	. . . . . . . . . . . . . . . 16 ⊢ (EXMID → ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔)
29	28	ad2antrr 485	. . . . . . . . . . . . . . 15 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔)
30		undifdcss 6900	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)) ↔ (ran 𝑔 ⊆ 𝑥 ∧ ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔))
31	26, 29, 30	sylanbrc 415	. . . . . . . . . . . . . 14 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → 𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)))
32	24, 31	eqtr4id 2222	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥)
33		df-fn 5201	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥)
35		rnun 5019	. . . . . . . . . . . . 13 ⊢ ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧}))
36		dfdm4 4803	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑔 = ran ◡𝑔
37		f1dm 5408	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → dom 𝑔 = 𝑦)
38	36, 37	eqtr3id 2217	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑦–1-1→𝑥 → ran ◡𝑔 = 𝑦)
39	38	uneq1d 3280	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑦–1-1→𝑥 → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})))
40		exmidexmid 4182	. . . . . . . . . . . . . . . . . 18 ⊢ (EXMID → DECID ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔))
41		exmiddc 831	. . . . . . . . . . . . . . . . . 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 5040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
44	43	adantr 274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
45		snssi 3724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦)
46	45	adantl 275	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → {𝑧} ⊆ 𝑦)
47	44, 46	eqsstrd 3183	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
48	47	ex 114	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
49		notm0 3435	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ↔ (𝑥 ∖ ran 𝑔) = ∅)
50		xpeq1 4625	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
51		0xp 4691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅ × {𝑧}) = ∅
52	50, 51	eqtrdi 2219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
53	52	rneqd 4840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ran ∅)
54		rn0 4867	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ran ∅ = ∅
55	53, 54	eqtrdi 2219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
56		0ss 3453	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ⊆ 𝑦
57	55, 56	eqsstrdi 3199	. . . . . . . . . . . . . . . . . . . 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 120	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧 ∈ 𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
60	48, 59	jaoi 711	. . . . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . . 15 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
63		ssequn2 3300	. . . . . . . . . . . . . . 15 ⊢ (ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦 ↔ (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
64	62, 63	sylib 121	. . . . . . . . . . . . . 14 ⊢ ((EXMID ∧ 𝑧 ∈ 𝑦) → (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
65	39, 64	sylan9eqr 2225	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
66	35, 65	eqtrid 2215	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
67		df-fo 5204	. . . . . . . . . . . 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 415	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦)
69		vex 2733	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
70	69	cnvex 5149	. . . . . . . . . . . . 13 ⊢ ◡𝑔 ∈ V
71		vex 2733	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
72		difexg 4130	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∖ ran 𝑔) ∈ V)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ ran 𝑔) ∈ V
74	5	snex 4171	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ V
75	73, 74	xpex 4726	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}) ∈ V
76	70, 75	unex 4426	. . . . . . . . . . . 12 ⊢ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∈ V
77		foeq1 5416	. . . . . . . . . . . 12 ⊢ (𝑓 = (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝑥–onto→𝑦 ↔ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦))
78	76, 77	spcev 2825	. . . . . . . . . . 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 563	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑔:𝑦–1-1→𝑥) ∧ 𝑧 ∈ 𝑦) → ∃𝑓 𝑓:𝑥–onto→𝑦)
81	80	ex 114	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑔:𝑦–1-1→𝑥) → (𝑧 ∈ 𝑦 → ∃𝑓 𝑓:𝑥–onto→𝑦))
82	81	exlimdv 1812	. . . . . . 7 ⊢ ((EXMID ∧ 𝑔:𝑦–1-1→𝑥) → (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑓 𝑓:𝑥–onto→𝑦))
83	82	imp 123	. . . . . 6 ⊢ (((EXMID ∧ 𝑔:𝑦–1-1→𝑥) ∧ ∃𝑧 𝑧 ∈ 𝑦) → ∃𝑓 𝑓:𝑥–onto→𝑦)
84	83	an32s 563	. . . . 5 ⊢ (((EXMID ∧ ∃𝑧 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
85	84	adantlrr 480	. . . 4 ⊢ (((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
86	2, 85	exlimddv 1891	. . 3 ⊢ ((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) → ∃𝑓 𝑓:𝑥–onto→𝑦)
87	86	ex 114	. 2 ⊢ (EXMID → ((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))
88	87	alrimivv 1868	1 ⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  Vcvv 2730   ∖ cdif 3118   ∪ cun 3119   ∩ cin 3120   ⊆ wss 3121  ∅c0 3414  {csn 3583   class class class wbr 3989  EXMIDwem 4180   × cxp 4609  ^◡ccnv 4610  dom cdm 4611  ran crn 4612  Fun wfun 5192   Fn wfn 5193  ⟶wf 5194  –1-1→wf1 5195  –onto→wfo 5196   ≼ cdom 6717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-exmid 4181  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-dom 6720

This theorem is referenced by:  exmidfodomr  7181