Theorem fodomfir 9363

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 9158). (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 8973	. . . . . . 7 ⊢ Rel ≺
2	1	brrelex2i 5731	. . . . . 6 ⊢ (∅ ≺ 𝐵 → 𝐵 ∈ V)
3		0sdomg 9134	. . . . . . 7 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅))
4		n0 4346	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
5	3, 4	bitrdi 286	. . . . . 6 ⊢ (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
6	2, 5	syl 17	. . . . 5 ⊢ (∅ ≺ 𝐵 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵))
7	6	ibi 266	. . . 4 ⊢ (∅ ≺ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
8		domfi 9219	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin)
9		simpl 481	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ Fin)
10		brdomi 8981	. . . . . . . 8 ⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴)
11		f1fn 6791	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔 Fn 𝐵)
12		fnfi 9208	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝐵 ∧ 𝐵 ∈ Fin) → 𝑔 ∈ Fin)
13	11, 12	sylan 578	. . . . . . . . . . 11 ⊢ ((𝑔:𝐵–1-1→𝐴 ∧ 𝐵 ∈ Fin) → 𝑔 ∈ Fin)
14	13	ex 411	. . . . . . . . . 10 ⊢ (𝑔:𝐵–1-1→𝐴 → (𝐵 ∈ Fin → 𝑔 ∈ Fin))
15		cnvfi 9207	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ Fin → ◡𝑔 ∈ Fin)
16		diffi 9206	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ ran 𝑔) ∈ Fin)
17		snfi 9073	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
18		xpfi 9353	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∖ ran 𝑔) ∈ Fin ∧ {𝑧} ∈ Fin) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
19	16, 17, 18	sylancl 584	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin)
20		unfi 9202	. . . . . . . . . . . . . 14 ⊢ ((◡𝑔 ∈ Fin ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ Fin) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
21	15, 19, 20	syl2an 594	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Fin ∧ 𝐴 ∈ Fin) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ Fin)
22		df-f1 6551	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
23	22	simprbi 495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔)
24		vex 3466	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
25	24	fconst 6780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
26		ffun 6723	. . . . . . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
29		df-rn 5685	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝑔 = dom ◡𝑔
30	29	eqcomi 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ dom ◡𝑔 = ran 𝑔
31	24	snnz 4775	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑧} ≠ ∅
32		dmxp 5927	. . . . . . . . . . . . . . . . . . . 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 4208	. . . . . . . . . . . . . . . . . 18 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
35		disjdif 4466	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
36	34, 35	eqtri 2754	. . . . . . . . . . . . . . . . 17 ⊢ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
37		funun 6597	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39	38	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
40		dmun 5909	. . . . . . . . . . . . . . . . . 18 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
41	29	uneq1i 4156	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
42	33	uneq2i 4157	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43	40, 41, 42	3eqtr2i 2760	. . . . . . . . . . . . . . . . 17 ⊢ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
44		f1f 6790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴)
45	44	frnd 6728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴)
46		undif 4476	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
47	45, 46	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
48	43, 47	eqtrid 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49	48	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
50		df-fn 6549	. . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
52		rnun 6149	. . . . . . . . . . . . . . 15 ⊢ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
53		dfdm4 5894	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝑔 = ran ◡𝑔
54		f1dm 6794	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵)
55	53, 54	eqtr3id 2780	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵)
56	55	uneq1d 4159	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
57		xpeq1 5688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
58		0xp 5772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {𝑧}) = ∅
59	57, 58	eqtrdi 2782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
60	59	rneqd 5936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
61		rn0 5924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran ∅ = ∅
62	60, 61	eqtrdi 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
63		0ss 4394	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ⊆ 𝐵
64	62, 63	eqsstrdi 4033	. . . . . . . . . . . . . . . . . . 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 6173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67	66	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
68		snssi 4807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵)
69	68	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
70	67, 69	eqsstrd 4017	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
71	70	ex 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
72	65, 71	pm2.61ine 3015	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
73		ssequn2 4181	. . . . . . . . . . . . . . . . 17 ⊢ (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
74	72, 73	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
75	56, 74	sylan9eqr 2788	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76	52, 75	eqtrid 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
77		df-fo 6552	. . . . . . . . . . . . . 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 581	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)
79		foeq1 6803	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵))
80	79	spcegv 3582	. . . . . . . . . . . . 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 415	. . . . . . . . . . 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 601	. . . . . . . . 9 ⊢ (𝑔:𝐵–1-1→𝐴 → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
85	84	exlimiv 1926	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)))
88	8, 9, 87	mp2and 697	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → (𝑧 ∈ 𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵))
89	88	exlimdv 1929	. . . 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 1114	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ ∅ ≺ 𝐵) → ∃𝑓 𝑓:𝐴–onto→𝐵)
92	91	3com23 1123	1 ⊢ ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930  Vcvv 3462   ∖ cdif 3943   ∪ cun 3944   ∩ cin 3945   ⊆ wss 3946  ∅c0 4322  {csn 4623   class class class wbr 5145   × cxp 5672  ^◡ccnv 5673  dom cdm 5674  ran crn 5675  Fun wfun 6540   Fn wfn 6541  ⟶wf 6542  –1-1→wf1 6543  –onto→wfo 6544   ≼ cdom 8964   ≺ csdm 8965  Fincfn 8966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-om 7869  df-1o 8488  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970

This theorem is referenced by:  fodomfib  9364