Theorem fodomfir 9366

Description: There exists a mapping from a finite set onto any nonempty set that it dominates, proved without using the Axiom of Power Sets (unlike fodomr 9167). (Contributed by BTernaryTau, 23-Jun-2025.)

Assertion

Ref	Expression
fodomfir	⊢ ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fodomfir

Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 8991	. . . . . . 7 ⊢ Rel ≺
2	1	brrelex2i 5746	. . . . . 6 ⊢ (∅ ≺ 𝐵 → 𝐵 ∈ V)
3		0sdomg 9143	. . . . . . 7 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
4		n0 4359	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
5	3, 4	bitrdi 287	. . . . . 6 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
6	2, 5	syl 17	. . . . 5 ⊢ (∅ ≺ 𝐵 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
7	6	ibi 267	. . . 4 ⊢ (∅ ≺ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
8		domfi 9227	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin)
9		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ Fin)
10		brdomi 8998	. . . . . . . 8 ⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴)
11		f1fn 6806	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔 Fn 𝐵)
12		fnfi 9216	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝐵 ∧ 𝐵 ∈ Fin) → 𝑔 ∈ Fin)
13	11, 12	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑔:𝐵–1-1→𝐴 ∧ 𝐵 ∈ Fin) → 𝑔 ∈ Fin)
14	13	ex 412	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (𝐵 ∈ Fin → 𝑔 ∈ Fin))
15		cnvfi 9215	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ Fin → ◡𝑔 ∈ Fin)
16		diffi 9214	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ ran 𝑔) ∈ Fin)
17		snfi 9082	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
18		xpfi 9356	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∖ ran 𝑔) ∈ Fin ∧ {𝑧} ∈ Fin) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
19	16, 17, 18	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
20		unfi 9210	. . . . . . . . . . . . . 14 ⊢ ((◡𝑔 ∈ Fin ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
21	15, 19, 20	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
22		df-f1 6568	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
23	22	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
24		vex 3482	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
25	24	fconst 6795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
26		ffun 6740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝐴 ∖ ran 𝑔) × {𝑧}))
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})
28	23, 27	jctir 520	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
29		df-rn 5700	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝑔 = dom ◡𝑔
30	29	eqcomi 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ dom ◡𝑔 = ran 𝑔
31	24	snnz 4781	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑧} ≠ ∅
32		dmxp 5942	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑧} ≠ ∅ → dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔))
33	31, 32	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔)
34	30, 33	ineq12i 4226	. . . . . . . . . . . . . . . . . 18 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
35		disjdif 4478	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
36	34, 35	eqtri 2763	. . . . . . . . . . . . . . . . 17 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
37		funun 6614	. . . . . . . . . . . . . . . . 17 ⊢ (((Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
38	28, 36, 37	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39	38	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
40		dmun 5924	. . . . . . . . . . . . . . . . . 18 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	29	uneq1i 4174	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
42	33	uneq2i 4175	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43	40, 41, 42	3eqtr2i 2769	. . . . . . . . . . . . . . . . 17 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
44		f1f 6805	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
45	44	frnd 6745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
46		undif 4488	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
47	45, 46	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
48	43, 47	eqtrid 2787	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49	48	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
50		df-fn 6566	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴))
51	39, 49, 50	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
52		rnun 6168	. . . . . . . . . . . . . . 15 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
53		dfdm4 5909	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝑔 = ran ◡𝑔
54		f1dm 6809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
55	53, 54	eqtr3id 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
56	55	uneq1d 4177	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
57		xpeq1 5703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
58		0xp 5787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {𝑧}) = ∅
59	57, 58	eqtrdi 2791	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
60	59	rneqd 5952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
61		rn0 5939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran ∅ = ∅
62	60, 61	eqtrdi 2791	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
63		0ss 4406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ⊆ 𝐵
64	62, 63	eqsstrdi 4050	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
65	64	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
66		rnxp 6192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67	66	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
68		snssi 4813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
69	68	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
70	67, 69	eqsstrd 4034	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
71	70	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
72	65, 71	pm2.61ine 3023	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
73		ssequn2 4199	. . . . . . . . . . . . . . . . 17 ⊢ (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
74	72, 73	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	56, 74	sylan9eqr 2797	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76	52, 75	eqtrid 2787	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
77		df-fo 6569	. . . . . . . . . . . . . 14 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 ↔ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵))
78	51, 76, 77	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
79		foeq1 6817	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
80	79	spcegv 3597	. . . . . . . . . . . . 13 ⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin → ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
81	21, 78, 80	syl2im 40	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵))
82	81	expcomd 416	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑔:𝐵–1-1→𝐴 → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
83	82	com12 32	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
84	14, 83	syland 603	. . . . . . . . 9 ⊢ (𝑔:𝐵–1-1→𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
85	84	exlimiv 1928	. . . . . . . 8 ⊢ (∃𝑔 𝑔:𝐵–1-1→𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
86	10, 85	syl 17	. . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
87	86	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
88	8, 9, 87	mp2and 699	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
89	88	exlimdv 1931	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
90	7, 89	syl5 34	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (∅ ≺ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
91	90	3impia 1116	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ ∅ ≺ 𝐵) → ∃𝑓 𝑓:𝐴–onto→𝐵)
92	91	3com23 1125	1 ⊢ ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631   class class class wbr 5148   × cxp 5687  ^◡ccnv 5688  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  –onto→wfo 6561   ≼ cdom 8982   ≺ csdm 8983  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988

This theorem is referenced by:  fodomfib  9367