Theorem exmidfodomrlemim 7057

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 6643	. . . . 5 ⊢ (𝑦 ≼ 𝑥 → ∃𝑔 𝑔:𝑦–1-1→𝑥)
2	1	ad2antll 482	. . . 4 ⊢ ((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) → ∃𝑔 𝑔:𝑦–1-1→𝑥)
3		df-f1 5128	. . . . . . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
6	5	fconst 5318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧}
7		ffun 5275	. . . . . . . . . . . . . . . . 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 4550	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑔 = dom ◡𝑔
11	10	eqcomi 2143	. . . . . . . . . . . . . . . . 17 ⊢ dom ◡𝑔 = ran 𝑔
12	5	snm 3643	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑤 𝑤 ∈ {𝑧}
13		dmxpm 4759	. . . . . . . . . . . . . . . . . 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 3275	. . . . . . . . . . . . . . . 16 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔))
16		disjdif 3435	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔)) = ∅
17	15, 16	eqtri 2160	. . . . . . . . . . . . . . 15 ⊢ (dom ◡𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅
18		funun 5167	. . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 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		f1rn 5329	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → ran 𝑔 ⊆ 𝑥)
22	21	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran 𝑔 ⊆ 𝑥)
23		exmidexmid 4120	. . . . . . . . . . . . . . . . 17 ⊢ (EXMID → DECID 𝑢 ∈ ran 𝑔)
24	23	ralrimivw 2506	. . . . . . . . . . . . . . . 16 ⊢ (EXMID → ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔)
25	24	ad2antrr 479	. . . . . . . . . . . . . . 15 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔)
26		undifdcss 6811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)) ↔ (ran 𝑔 ⊆ 𝑥 ∧ ∀𝑢 ∈ 𝑥 DECID 𝑢 ∈ ran 𝑔))
27	22, 25, 26	sylanbrc 413	. . . . . . . . . . . . . 14 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → 𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)))
28		dmun 4746	. . . . . . . . . . . . . . 15 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
29	10	uneq1i 3226	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
30	14	uneq2i 3227	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
31	28, 29, 30	3eqtr2i 2166	. . . . . . . . . . . . . 14 ⊢ dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
32	27, 31	syl6reqr 2191	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → dom (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥)
33		df-fn 5126	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥)
35		rnun 4947	. . . . . . . . . . . . 13 ⊢ ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧}))
36		dfdm4 4731	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑔 = ran ◡𝑔
37		f1dm 5333	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝑦–1-1→𝑥 → dom 𝑔 = 𝑦)
38	36, 37	syl5eqr 2186	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑦–1-1→𝑥 → ran ◡𝑔 = 𝑦)
39	38	uneq1d 3229	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑦–1-1→𝑥 → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})))
40		exmidexmid 4120	. . . . . . . . . . . . . . . . . 18 ⊢ (EXMID → DECID ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔))
41		exmiddc 821	. . . . . . . . . . . . . . . . . 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 4968	. . . . . . . . . . . . . . . . . . . . 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 3664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦)
46	45	adantl 275	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧 ∈ 𝑦) → {𝑧} ⊆ 𝑦)
47	44, 46	eqsstrd 3133	. . . . . . . . . . . . . . . . . . 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 3383	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ↔ (𝑥 ∖ ran 𝑔) = ∅)
50		xpeq1 4553	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
51		0xp 4619	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅ × {𝑧}) = ∅
52	50, 51	syl6eq 2188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
53	52	rneqd 4768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ran ∅)
54		rn0 4795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ran ∅ = ∅
55	53, 54	syl6eq 2188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
56		0ss 3401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ⊆ 𝑦
57	55, 56	eqsstrdi 3149	. . . . . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . . . 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 3249	. . . . . . . . . . . . . . 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 2194	. . . . . . . . . . . . 13 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (ran ◡𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
66	35, 65	syl5eq 2184	. . . . . . . . . . . 12 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ran (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
67		df-fo 5129	. . . . . . . . . . . 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 413	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦)
69		vex 2689	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
70	69	cnvex 5077	. . . . . . . . . . . . 13 ⊢ ◡𝑔 ∈ V
71		vex 2689	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
72		difexg 4069	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∖ ran 𝑔) ∈ V)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∖ ran 𝑔) ∈ V
74	5	snex 4109	. . . . . . . . . . . . . 14 ⊢ {𝑧} ∈ V
75	73, 74	xpex 4654	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ ran 𝑔) × {𝑧}) ∈ V
76	70, 75	unex 4362	. . . . . . . . . . . 12 ⊢ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∈ V
77		foeq1 5341	. . . . . . . . . . . 12 ⊢ (𝑓 = (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝑥–onto→𝑦 ↔ (◡𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥–onto→𝑦))
78	76, 77	spcev 2780	. . . . . . . . . . 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 557	. . . . . . . . 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 1791	. . . . . . 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 557	. . . . 5 ⊢ (((EXMID ∧ ∃𝑧 𝑧 ∈ 𝑦) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
85	84	adantlrr 474	. . . 4 ⊢ (((EXMID ∧ (∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥)) ∧ 𝑔:𝑦–1-1→𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)
86	2, 85	exlimddv 1870	. . 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 1847	1 ⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ∖ cdif 3068   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  {csn 3527   class class class wbr 3929  EXMIDwem 4118   × cxp 4537  ^◡ccnv 4538  dom cdm 4539  ran crn 4540  Fun wfun 5117   Fn wfn 5118  ⟶wf 5119  –1-1→wf1 5120  –onto→wfo 5121   ≼ cdom 6633

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-exmid 4119  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-dom 6636

This theorem is referenced by:  exmidfodomr  7060