Theorem ufldom 23986

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 9002	. . 3 ⊢ (𝑋 ∈ UFL → (𝑌 ≼ 𝑋 ↔ ∃𝑥(𝑌 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑋)))
2		bren 8994	. . . . . . . 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 23942	. . . . . . 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 23872	. . . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → 𝑓:𝑌⟶𝑥)
9	8	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → 𝑓:𝑌⟶𝑥)
10		fmfil 23968	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ UFL ∧ 𝑔 ∈ (fBas‘𝑌) ∧ 𝑓:𝑌⟶𝑥) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
11	5, 7, 9, 10	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥))
12		ufli 23938	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ UFL ∧ ((𝑥 FilMap 𝑓)‘𝑔) ∈ (Fil‘𝑥)) → ∃𝑦 ∈ (UFil‘𝑥)((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
13	5, 11, 12	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ∃𝑦 ∈ (UFil‘𝑥)((𝑥 FilMap 𝑓)‘𝑔) ⊆ 𝑦)
14		f1odm 6853	. . . . . . . . . . . . . . . . . 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 3482	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
17	16	dmex 7932	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
18	15, 17	eqeltrrdi 2848	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → 𝑌 ∈ V)
19	18	ad2antrr 726	. . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → ◡𝑓:𝑥–1-1-onto→𝑌)
22	21	ad3antrrr 730	. . . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . . . 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 23983	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ V ∧ 𝑦 ∈ (UFil‘𝑥) ∧ ◡𝑓:𝑥⟶𝑌) → ((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌))
26	19, 20, 24, 25	syl3anc 1370	. . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝑌–1-1-onto→𝑥 → (◡𝑓 ∘ 𝑓) = ( I ↾ 𝑌))
28	27	ad3antrrr 730	. . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 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 23985	. . . . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . . . 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 23984	. . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . 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 23872	. . . . . . . . . . . . . . . . 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 23928	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (UFil‘𝑥) → 𝑦 ∈ (Fil‘𝑥))
44		filfbas 23872	. . . . . . . . . . . . . . . . 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 23970	. . . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . . 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 4035	. . . . . . . . . . . . . 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 4022	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ((𝑌 FilMap ◡𝑓)‘𝑦) → (𝑔 ⊆ 𝑢 ↔ 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦)))
51	50	rspcev 3622	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 FilMap ◡𝑓)‘𝑦) ∈ (UFil‘𝑌) ∧ 𝑔 ⊆ ((𝑌 FilMap ◡𝑓)‘𝑦)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
52	26, 49, 51	syl2anc 584	. . . . . . . . . . . . 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 3159	. . . . . . . . . . . 12 ⊢ (((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) ∧ 𝑔 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
54	53	ralrimiva 3144	. . . . . . . . . . 11 ⊢ ((𝑓:𝑌–1-1-onto→𝑥 ∧ 𝑥 ∈ UFL) → ∀𝑔 ∈ (Fil‘𝑌)∃𝑢 ∈ (UFil‘𝑌)𝑔 ⊆ 𝑢)
55		isufl 23937	. . . . . . . . . . . 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 1928	. . . . . . . 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 596	. . . . . 6 ⊢ ((𝑌 ≈ 𝑥 ∧ (𝑋 ∈ UFL ∧ 𝑥 ⊆ 𝑋)) → 𝑌 ∈ UFL)
62	61	an12s 649	. . . . 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 1931	. . 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 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963   class class class wbr 5148   I cid 5582  ^◡ccnv 5688  dom cdm 5689   ↾ cres 5691   ∘ ccom 5693  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ≈ cen 8981   ≼ cdom 8982  fBascfbas 21370  Filcfil 23869  UFilcufil 23923  UFLcufl 23924   FilMap cfm 23957

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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  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-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-int 4952  df-iun 4998  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-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925  df-ufl 23926  df-fm 23962

This theorem is referenced by: (None)