Theorem ufldom 23940

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 8903	. . 3 ⊢ (𝑋 ∈ UFL → (𝑌 ≼ 𝑋 ↔ ∃𝑥(𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋)))
2		bren 8897	. . . . . . . 8 ⊢ (𝑌 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝑌–1-1-onto→𝑥)
3	2	biimpi 216	. . . . . . 7 ⊢ (𝑌 ≈ 𝑥 → ∃𝑓 𝑓:𝑌–1-1-onto→𝑥)
4		ssufl 23896	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ UFL)
5		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑥 ∈ UFL)
6		filfbas 23826	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ (Fil‘𝑌) → 𝑔 ∈ (fBas‘𝑌))
7	6	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑔 ∈ (fBas‘𝑌))
8		f1of 6775	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → 𝑓:𝑌⟶𝑥)
9	8	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑓:𝑌⟶𝑥)
10		fmfil 23922	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ UFL ∧ 𝑔 ∈ (fBas‘𝑌) ∧ 𝑓:𝑌⟶𝑥) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
11	5, 7, 9, 10	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
12		ufli 23892	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ UFL ∧ ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥)) → ∃𝑦 ∈ (UFil‘𝑥)((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
13	5, 11, 12	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ∃𝑦 ∈ (UFil‘𝑥)((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
14		f1odm 6779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → dom 𝑓 = 𝑌)
15	14	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → dom 𝑓 = 𝑌)
16		vex 3434	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
17	16	dmex 7854	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
18	15, 17	eqeltrrdi 2846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ V)
19	18	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) ∧ (𝑦 ∈ (UFil‘𝑥) ∧ ((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)) → 𝑌 ∈ V)
20		simprl 771	. . . . . . . . . . . . . . 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 6787	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → ◡𝑓:𝑥–1-1-onto→𝑌)
22	21	ad3antrrr 731	. . . . . . . . . . . . . . . 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 6775	. . . . . . . . . . . . . . . 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 23937	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ V ∧ 𝑦 ∈ (UFil‘𝑥) ∧ ◡𝑓:𝑥⟶𝑌) → ((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌))
26	19, 20, 24, 25	syl3anc 1374	. . . . . . . . . . . . . 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 6804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → (◡𝑓 ∘ 𝑓) = ( I ↾ 𝑌))
28	27	ad3antrrr 731	. . . . . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . . . 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 6837	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . 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 23939	. . . . . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 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 23938	. . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 23826	. . . . . . . . . . . . . . . . 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 23882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (UFil‘𝑥) → 𝑦 ∈ (Fil‘𝑥))
44		filfbas 23826	. . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . 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 23924	. . . . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ((𝑌 FilMap ◡𝑓)‘𝑦) → (𝑔 ⊆ 𝑢 ↔ 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦)))
51	50	rspcev 3565	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌) ∧ 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
52	26, 49, 51	syl2anc 585	. . . . . . . . . . . . 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 3145	. . . . . . . . . . . 12 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
54	53	ralrimiva 3130	. . . . . . . . . . 11 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → ∀𝑔 ∈ (Fil‘𝑌)∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
55		isufl 23891	. . . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ UFL)
58	57	ex 412	. . . . . . . . 9 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → (𝑥 ∈ UFL → 𝑌 ∈ UFL))
59	58	exlimiv 1932	. . . . . . . 8 ⊢ (∃𝑓 𝑓:𝑌–1-1-onto→𝑥 → (𝑥 ∈ UFL → 𝑌 ∈ UFL))
60	59	imp 406	. . . . . . 7 ⊢ ((∃𝑓 𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ UFL)
61	3, 4, 60	syl2an 597	. . . . . 6 ⊢ ((𝑌 ≈ 𝑥 ∧ (𝑋 ∈ UFL ∧ 𝑥 ⊆ 𝑋)) → 𝑌 ∈ UFL)
62	61	an12s 650	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ (𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋)) → 𝑌 ∈ UFL)
63	62	ex 412	. . . 4 ⊢ (𝑋 ∈ UFL → ((𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋) → 𝑌 ∈ UFL))
64	63	exlimdv 1935	. . 3 ⊢ (𝑋 ∈ UFL → (∃𝑥(𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋) → 𝑌 ∈ UFL))
65	1, 64	sylbid 240	. 2 ⊢ (𝑋 ∈ UFL → (𝑌 ≼ 𝑋 → 𝑌 ∈ UFL))
66	65	imp 406	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ≼ 𝑋) → 𝑌 ∈ UFL)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086   I cid 5519  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   ∘ ccom 5629  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361   ≈ cen 8884   ≼ cdom 8885  fBascfbas 21335  Filcfil 23823  UFilcufil 23877  UFLcufl 23878   FilMap cfm 23911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-1o 8399  df-2o 8400  df-en 8888  df-dom 8889  df-fin 8891  df-fi 9318  df-rest 17379  df-fbas 21344  df-fg 21345  df-fil 23824  df-ufil 23879  df-ufl 23880  df-fm 23916

This theorem is referenced by: (None)