Theorem ufldom 24023

Description: The ultrafilter lemma property is a cardinal invariant, so since it transfers to subsets it also transfers over set dominance. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
ufldom	⊢ ((𝑋 ∈ UFL ∧ 𝑌 ≼ 𝑋) → 𝑌 ∈ UFL)

Proof of Theorem ufldom

Dummy variables 𝑢 𝑥 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		domeng 8944	. . 3 ⊢ (𝑋 ∈ UFL → (𝑌 ≼ 𝑋 ↔ ∃𝑥(𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋)))
2		bren 8938	. . . . . . . 8 ⊢ (𝑌 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝑌–1-1-onto→𝑥)
3	2	biimpi 218	. . . . . . 7 ⊢ (𝑌 ≈ 𝑥 → ∃𝑓 𝑓:𝑌–1-1-onto→𝑥)
4		ssufl 23979	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ UFL)
5		simplr 778	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑥 ∈ UFL)
6		filfbas 23909	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ (Fil‘𝑌) → 𝑔 ∈ (fBas‘𝑌))
7	6	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑔 ∈ (fBas‘𝑌))
8		f1of 6807	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → 𝑓:𝑌⟶𝑥)
9	8	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑓:𝑌⟶𝑥)
10		fmfil 24005	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ UFL ∧ 𝑔 ∈ (fBas‘𝑌) ∧ 𝑓:𝑌⟶𝑥) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
11	5, 7, 9, 10	syl3anc 1391	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
12		ufli 23975	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ UFL ∧ ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥)) → ∃𝑦 ∈ (UFil‘𝑥)((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
13	5, 11, 12	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ∃𝑦 ∈ (UFil‘𝑥)((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
14		f1odm 6811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → dom 𝑓 = 𝑌)
15	14	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → dom 𝑓 = 𝑌)
16		vex 3459	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
17	16	dmex 7891	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
18	15, 17	eqeltrrdi 2872	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ V)
19	18	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑌 ∈ V)
20		simprl 780	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑦 ∈ (UFil‘𝑥))
21		f1ocnv 6820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → ◡𝑓:𝑥–1-1-onto→𝑌)
22	21	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ◡𝑓:𝑥–1-1-onto→𝑌)
23		f1of 6807	. . . . . . . . . . . . . . . 16 ⊢ (◡𝑓:𝑥–1-1-onto→𝑌 → ◡𝑓:𝑥⟶𝑌)
24	22, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ◡𝑓:𝑥⟶𝑌)
25		fmufil 24020	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ V ∧ 𝑦 ∈ (UFil‘𝑥) ∧ ◡𝑓:𝑥⟶𝑌) → ((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌))
26	19, 20, 24, 25	syl3anc 1391	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌))
27		f1ococnv1 6837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → (◡𝑓 ∘ 𝑓) = ( I ↾ 𝑌))
28	27	ad3antrrr 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → (◡𝑓 ∘ 𝑓) = ( I ↾ 𝑌))
29	28	oveq2d 7413	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → (𝑌 FilMap (◡𝑓 ∘ 𝑓)) = (𝑌 FilMap ( I ↾ 𝑌)))
30	29	fveq1d 6870	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap (◡𝑓 ∘ 𝑓))‘𝑔) = ((𝑌 FilMap ( I ↾ 𝑌))‘𝑔))
31	5	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑥 ∈ UFL)
32	7	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑔 ∈ (fBas‘𝑌))
33	8	ad3antrrr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑓:𝑌⟶𝑥)
34		fmco 24022	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑌 ∈ V ∧ 𝑥 ∈ UFL ∧ 𝑔 ∈ (fBas‘𝑌)) ∧ (◡𝑓:𝑥⟶𝑌 ∧ 𝑓:𝑌⟶𝑥)) → ((𝑌 FilMap (◡𝑓 ∘ 𝑓))‘𝑔) = ((𝑌 FilMap ◡𝑓)‘((𝑥 FilMap 𝑓)‘𝑔)))
35	19, 31, 32, 24, 33, 34	syl32anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap (◡𝑓 ∘ 𝑓))‘𝑔) = ((𝑌 FilMap ◡𝑓)‘((𝑥 FilMap 𝑓)‘𝑔)))
36		simplr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑔 ∈ (Fil‘𝑌))
37		fmid 24021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ (Fil‘𝑌) → ((𝑌 FilMap ( I ↾ 𝑌))‘𝑔) = 𝑔)
38	36, 37	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap ( I ↾ 𝑌))‘𝑔) = 𝑔)
39	30, 35, 38	3eqtr3d 2806	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap ◡𝑓)‘((𝑥 FilMap 𝑓)‘𝑔)) = 𝑔)
40	11	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
41		filfbas 23909	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (fBas‘𝑥))
42	40, 41	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (fBas‘𝑥))
43		ufilfil 23965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (UFil‘𝑥) → 𝑦 ∈ (Fil‘𝑥))
44		filfbas 23909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (Fil‘𝑥) → 𝑦 ∈ (fBas‘𝑥))
45	20, 43, 44	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑦 ∈ (fBas‘𝑥))
46		simprr 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
47		fmss 24007	. . . . . . . . . . . . . . . 16 ⊢ (((𝑌 ∈ V ∧ ((𝑥 FilMap 𝑓)‘𝑔) ∈ (fBas‘𝑥) ∧ 𝑦 ∈ (fBas‘𝑥)) ∧ (◡𝑓:𝑥⟶𝑌 ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap ◡𝑓)‘((𝑥 FilMap 𝑓)‘𝑔)) ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦))
48	19, 42, 45, 24, 46, 47	syl32anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ((𝑌 FilMap ◡𝑓)‘((𝑥 FilMap 𝑓)‘𝑔)) ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦))
49	39, 48	eqsstrrd 3972	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦))
50		sseq2 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ((𝑌 FilMap ◡𝑓)‘𝑦) → (𝑔 ⊆ 𝑢 ↔ 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦)))
51	50	rspcev 3582	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌) ∧ 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
52	26, 49, 51	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
53	13, 52	rexlimddv 3170	. . . . . . . . . . . 12 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
54	53	ralrimiva 3155	. . . . . . . . . . 11 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → ∀𝑔 ∈ (Fil‘𝑌)∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
55		isufl 23974	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ V → (𝑌 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑌)∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢))
56	18, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → (𝑌 ∈ UFL ↔ ∀𝑔 ∈ (Fil‘𝑌)∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢))
57	54, 56	mpbird 259	. . . . . . . . . 10 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ UFL)
58	57	ex 416	. . . . . . . . 9 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → (𝑥 ∈ UFL → 𝑌 ∈ UFL))
59	58	exlimiv 1951	. . . . . . . 8 ⊢ (∃𝑓 𝑓:𝑌–1-1-onto→𝑥 → (𝑥 ∈ UFL → 𝑌 ∈ UFL))
60	59	imp 410	. . . . . . 7 ⊢ ((∃𝑓 𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ UFL)
61	3, 4, 60	syl2an 605	. . . . . 6 ⊢ ((𝑌 ≈ 𝑥 ∧ (𝑋 ∈ UFL ∧ 𝑥 ⊆ 𝑋)) → 𝑌 ∈ UFL)
62	61	an12s 659	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ (𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋)) → 𝑌 ∈ UFL)
63	62	ex 416	. . . 4 ⊢ (𝑋 ∈ UFL → ((𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋) → 𝑌 ∈ UFL))
64	63	exlimdv 1954	. . 3 ⊢ (𝑋 ∈ UFL → (∃𝑥(𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋) → 𝑌 ∈ UFL))
65	1, 64	sylbid 242	. 2 ⊢ (𝑋 ∈ UFL → (𝑌 ≼ 𝑋 → 𝑌 ∈ UFL))
66	65	imp 410	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ≼ 𝑋) → 𝑌 ∈ UFL)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087  Vcvv 3455   ⊆ wss 3905   class class class wbr 5101   I cid 5542  ^◡ccnv 5647  dom cdm 5648   ↾ cres 5650   ∘ ccom 5652  ⟶wf 6518  –1-1-onto→wf1o 6521  ‘cfv 6522  (class class class)co 7397   ≈ cen 8925   ≼ cdom 8926  fBascfbas 21413  Filcfil 23906  UFilcufil 23960  UFLcufl 23961   FilMap cfm 23994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-1o 8438  df-2o 8439  df-en 8929  df-dom 8930  df-fin 8932  df-fi 9358  df-rest 17452  df-fbas 21422  df-fg 21423  df-fil 23907  df-ufil 23962  df-ufl 23963  df-fm 23999

This theorem is referenced by: (None)