Theorem exmidfodomrlemim 7412

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 6920	. . . . 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 5331	. . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
6	5	fconst 5532	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧}
7		ffun 5485	. . . . . . . . . . . . . . . . 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 4736	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑔 = dom ◡𝑔
11	10	eqcomi 2235	. . . . . . . . . . . . . . . . 17 ⊢ dom ◡𝑔 = ran 𝑔
12	5	snm 3792	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑤 𝑤 ∈ {𝑧}
13		dmxpm 4952	. . . . . . . . . . . . . . . . . 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 3406	. . . . . . . . . . . . . . . 16 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔))
16		disjdif 3567	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔)) = ∅
17	15, 16	eqtri 2252	. . . . . . . . . . . . . . 15 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅
18		funun 5371	. . . . . . . . . . . . . . 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 4938	. . . . . . . . . . . . . . 15 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
22	10	uneq1i 3357	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
23	14	uneq2i 3358	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
24	21, 22, 23	3eqtr2i 2258	. . . . . . . . . . . . . 14 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
25		f1rn 5543	. . . . . . . . . . . . . . . 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 4286	. . . . . . . . . . . . . . . . 17 ⊢ (EXMID → DECID 𝑢 ∈ ran 𝑔)
28	27	ralrimivw 2606	. . . . . . . . . . . . . . . 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 7115	. . . . . . . . . . . . . . 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 2283	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥)
33		df-fn 5329	. . . . . . . . . . . . 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 5145	. . . . . . . . . . . . 13 ⊢ ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧}))
36		dfdm4 4923	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑔 = ran ◡𝑔
37		f1dm 5547	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → dom 𝑔 = 𝑦)
38	36, 37	eqtr3id 2278	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑦–1-1→𝑥 → ran ◡𝑔 = 𝑦)
39	38	uneq1d 3360	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑦–1-1→𝑥 → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})))
40		exmidexmid 4286	. . . . . . . . . . . . . . . . . 18 ⊢ (EXMID → DECID ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔))
41		exmiddc 843	. . . . . . . . . . . . . . . . . 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 5166	. . . . . . . . . . . . . . . . . . . . 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 3817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦)
46	45	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → {𝑧} ⊆ 𝑦)
47	44, 46	eqsstrd 3263	. . . . . . . . . . . . . . . . . . 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 3515	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ↔ (𝑥 ∖ ran 𝑔) = ∅)
50		xpeq1 4739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
51		0xp 4806	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅ × {𝑧}) = ∅
52	50, 51	eqtrdi 2280	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
53	52	rneqd 4961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ran ∅)
54		rn0 4988	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ran ∅ = ∅
55	53, 54	eqtrdi 2280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
56		0ss 3533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ⊆ 𝑦
57	55, 56	eqsstrdi 3279	. . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 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 3380	. . . . . . . . . . . . . . 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 2286	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
66	35, 65	eqtrid 2276	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
67		df-fo 5332	. . . . . . . . . . . 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 2805	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
70	69	cnvex 5275	. . . . . . . . . . . . 13 ⊢ ◡𝑔 ∈ V
71		vex 2805	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
72		difexg 4231	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∖ ran 𝑔) ∈ V)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ ran 𝑔) ∈ V
74	5	snex 4275	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ V
75	73, 74	xpex 4842	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}) ∈ V
76	70, 75	unex 4538	. . . . . . . . . . . 12 ⊢ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∈ V
77		foeq1 5555	. . . . . . . . . . . 12 ⊢ (𝑓 = (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝑥–onto→𝑦 ↔ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦))
78	76, 77	spcev 2901	. . . . . . . . . . 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 570	. . . . . . . . 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 1867	. . . . . . 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 570	. . . . 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 1947	. . 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 1923	1 ⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715  DECID wdc 841  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  {csn 3669   class class class wbr 4088  EXMIDwem 4284   × cxp 4723  ^◡ccnv 4724  dom cdm 4725  ran crn 4726  Fun wfun 5320   Fn wfn 5321  ⟶wf 5322  –1-1→wf1 5323  –onto→wfo 5324   ≼ cdom 6908

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-exmid 4285  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-dom 6911

This theorem is referenced by:  exmidfodomr  7415