Theorem fodomfir 9294

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 9123). (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 8956	. . . . . . 7 ⊢ Rel ≺
2	1	brrelex2i 5720	. . . . . 6 ⊢ (∅ ≺ 𝐵 → 𝐵 ∈ V)
3		0sdomg 9101	. . . . . . 7 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
4		n0 4307	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
5	3, 4	bitrdi 290	. . . . . 6 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
6	2, 5	syl 18	. . . . 5 ⊢ (∅ ≺ 𝐵 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
7	6	ibi 270	. . . 4 ⊢ (∅ ≺ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
8		domfi 9180	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin)
9		simpl 488	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ Fin)
10		brdomi 8962	. . . . . . . 8 ⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴)
11		f1fn 6779	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔 Fn 𝐵)
12		fnfi 9169	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝐵 ∧ 𝐵 ∈ Fin) → 𝑔 ∈ Fin)
13	11, 12	sylan 592	. . . . . . . . . . 11 ⊢ ((𝑔:𝐵–1-1→𝐴 ∧ 𝐵 ∈ Fin) → 𝑔 ∈ Fin)
14	13	ex 418	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (𝐵 ∈ Fin → 𝑔 ∈ Fin))
15		cnvfi 9167	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ Fin → ◡𝑔 ∈ Fin)
16		diffi 9166	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ ran 𝑔) ∈ Fin)
17		snfi 9047	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
18		xpfi 9286	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∖ ran 𝑔) ∈ Fin ∧ {𝑧} ∈ Fin) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
19	16, 17, 18	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
20		unfi 9162	. . . . . . . . . . . . . 14 ⊢ ((◡𝑔 ∈ Fin ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
21	15, 19, 20	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
22		df-f1 6545	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
23	22	simprbi 503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
24		vex 3461	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
25	24	fconst 6768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
26		ffun 6712	. . . . . . . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
29		df-rn 5674	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝑔 = dom ◡𝑔
30	29	eqcomi 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ dom ◡𝑔 = ran 𝑔
31	24	snnz 4744	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑧} ≠ ∅
32		dmxp 5921	. . . . . . . . . . . . . . . . . . . 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 4171	. . . . . . . . . . . . . . . . . 18 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
35		disjdif 4433	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
36	34, 35	eqtri 2788	. . . . . . . . . . . . . . . . 17 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
37		funun 6586	. . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39	38	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
40		dmun 5902	. . . . . . . . . . . . . . . . . 18 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	29	uneq1i 4118	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
42	33	uneq2i 4119	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43	40, 41, 42	3eqtr2i 2794	. . . . . . . . . . . . . . . . 17 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
44		f1f 6778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
45	44	frnd 6718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
46		undif 4445	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
47	45, 46	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
48	43, 47	eqtrid 2812	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49	48	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
50		df-fn 6543	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
52		rnun 6144	. . . . . . . . . . . . . . 15 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
53		dfdm4 5887	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝑔 = ran ◡𝑔
54		f1dm 6784	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
55	53, 54	eqtr3id 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
56	55	uneq1d 4121	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
57		xpeq1 5677	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
58		0xp 5762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {𝑧}) = ∅
59	57, 58	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
60	59	rneqd 5930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
61		rn0 5918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran ∅ = ∅
62	60, 61	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
63		0ss 4357	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ⊆ 𝐵
64	62, 63	eqsstrdi 3982	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
65	64	a1d 26	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
66		rnxp 6170	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67	66	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
68		snssi 4753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
69	68	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
70	67, 69	eqsstrd 3972	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
71	70	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
72	65, 71	pm2.61ine 3043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
73		ssequn2 4142	. . . . . . . . . . . . . . . . 17 ⊢ (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
74	72, 73	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	56, 74	sylan9eqr 2822	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76	52, 75	eqtrid 2812	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
77		df-fo 6546	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
79		foeq1 6792	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
80	79	spcegv 3558	. . . . . . . . . . . . 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 41	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵))
82	81	expcomd 422	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑔:𝐵–1-1→𝐴 → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
83	82	com12 33	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
84	14, 83	syland 615	. . . . . . . . 9 ⊢ (𝑔:𝐵–1-1→𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
85	84	exlimiv 1963	. . . . . . . 8 ⊢ (∃𝑔 𝑔:𝐵–1-1→𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
86	10, 85	syl 18	. . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
87	86	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
88	8, 9, 87	mp2and 712	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
89	88	exlimdv 1966	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
90	7, 89	syl5 35	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (∅ ≺ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
91	90	3impia 1135	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ ∅ ≺ 𝐵) → ∃𝑓 𝑓:𝐴–onto→𝐵)
92	91	3com23 1144	1 ⊢ ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591   class class class wbr 5111   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  ran crn 5664  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537  –onto→wfo 6538   ≼ cdom 8947   ≺ csdm 8948  Fincfn 8949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7869  df-1o 8459  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953

This theorem is used by:  fodomfib  9295