Theorem cvmlift2lem9a 36037

Description: Lemma for cvmlift2 36050 and cvmlift3 36062. (Contributed by Mario Carneiro, 9-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift2lem9a.b	⊢ 𝐵 = ∪ 𝐶
cvmlift2lem9a.y	⊢ 𝑌 = ∪ 𝐾
cvmlift2lem9a.s	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})
cvmlift2lem9a.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift2lem9a.h	⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
cvmlift2lem9a.g	⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
cvmlift2lem9a.k	⊢ (𝜑 → 𝐾 ∈ Top)
cvmlift2lem9a.1	⊢ (𝜑 → 𝑋 ∈ 𝑌)
cvmlift2lem9a.2	⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
cvmlift2lem9a.3	⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
cvmlift2lem9a.4	⊢ (𝜑 → 𝑀 ⊆ 𝑌)
cvmlift2lem9a.6	⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊)

Assertion

Ref	Expression
cvmlift2lem9a	⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))

Distinct variable groups:   𝑐,𝑑,𝑘,𝑠,𝐴   𝐹,𝑐,𝑑,𝑘,𝑠   𝐽,𝑐,𝑑,𝑘,𝑠   𝑇,𝑐,𝑑,𝑠   𝐶,𝑐,𝑑,𝑘,𝑠   𝑊,𝑐,𝑑

Allowed substitution hints:   𝜑(𝑘, 𝑠, 𝑐, 𝑑)   𝐵(𝑘, 𝑠, 𝑐, 𝑑)   𝑆(𝑘, 𝑠, 𝑐, 𝑑)   𝑇(𝑘)   𝐻(𝑘, 𝑠, 𝑐, 𝑑)   𝐾(𝑘, 𝑠, 𝑐, 𝑑)   𝑀(𝑘, 𝑠, 𝑐, 𝑑)   𝑊(𝑘, 𝑠)   𝑋(𝑘, 𝑠, 𝑐, 𝑑)   𝑌(𝑘, 𝑠, 𝑐, 𝑑)

Proof of Theorem cvmlift2lem9a

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2lem9a.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
2		cvmtop1 35994	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝐶 ∈ Top)
4		cnrest2r 23585	. . 3 ⊢ (𝐶 ∈ Top → ((𝐾 ↾t 𝑀) Cn (𝐶 ↾t 𝑊)) ⊆ ((𝐾 ↾t 𝑀) Cn 𝐶))
5	3, 4	syl 18	. 2 ⊢ (𝜑 → ((𝐾 ↾t 𝑀) Cn (𝐶 ↾t 𝑊)) ⊆ ((𝐾 ↾t 𝑀) Cn 𝐶))
6		cvmlift2lem9a.h	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
7	6	ffnd 6702	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝑌)
8		cvmlift2lem9a.4	. . . . 5 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
9		fnssres 6654	. . . . 5 ⊢ ((𝐻 Fn 𝑌 ∧ 𝑀 ⊆ 𝑌) → (𝐻 ↾ 𝑀) Fn 𝑀)
10	7, 8, 9	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐻 ↾ 𝑀) Fn 𝑀)
11		df-ima 5664	. . . . 5 ⊢ (𝐻 “ 𝑀) = ran (𝐻 ↾ 𝑀)
12		cvmlift2lem9a.6	. . . . 5 ⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊)
13	11, 12	eqsstrrid 3970	. . . 4 ⊢ (𝜑 → ran (𝐻 ↾ 𝑀) ⊆ 𝑊)
14		df-f 6535	. . . 4 ⊢ ((𝐻 ↾ 𝑀):𝑀⟶𝑊 ↔ ((𝐻 ↾ 𝑀) Fn 𝑀 ∧ ran (𝐻 ↾ 𝑀) ⊆ 𝑊))
15	10, 13, 14	sylanbrc 595	. . 3 ⊢ (𝜑 → (𝐻 ↾ 𝑀):𝑀⟶𝑊)
16		cvmlift2lem9a.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
17		cvmlift2lem9a.3	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
18	17	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ 𝑇)
19		cvmlift2lem9a.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})
20	19	cvmsf1o 36006	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝐴) ∧ 𝑊 ∈ 𝑇) → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝐴)
21	1, 16, 18, 20	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝐴)
22	21	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝐴)
23		f1of1 6815	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝑊):𝑊–1-1-onto→𝐴 → (𝐹 ↾ 𝑊):𝑊–1-1→𝐴)
24	22, 23	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (𝐹 ↾ 𝑊):𝑊–1-1→𝐴)
25		cvmlift2lem9a.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ∪ 𝐶
26	25	toptopon 23215	. . . . . . . . . . 11 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
27	3, 26	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
28	19	cvmsss 36001	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ (𝑆‘𝐴) → 𝑇 ⊆ 𝐶)
29	16, 28	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ⊆ 𝐶)
30	29, 18	sseldd 3932	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
31		toponss 23225	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊 ∈ 𝐶) → 𝑊 ⊆ 𝐵)
32	27, 30, 31	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ 𝐵)
33		resttopon 23459	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊 ⊆ 𝐵) → (𝐶 ↾t 𝑊) ∈ (TopOn‘𝑊))
34	27, 32, 33	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ↾t 𝑊) ∈ (TopOn‘𝑊))
35		toponss 23225	. . . . . . . . 9 ⊢ (((𝐶 ↾t 𝑊) ∈ (TopOn‘𝑊) ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → 𝑥 ⊆ 𝑊)
36	34, 35	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → 𝑥 ⊆ 𝑊)
37		f1imacnv 6833	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝑊):𝑊–1-1→𝐴 ∧ 𝑥 ⊆ 𝑊) → (◡(𝐹 ↾ 𝑊) “ ((𝐹 ↾ 𝑊) “ 𝑥)) = 𝑥)
38	24, 36, 37	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡(𝐹 ↾ 𝑊) “ ((𝐹 ↾ 𝑊) “ 𝑥)) = 𝑥)
39	38	imaeq2d 6054	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡(𝐻 ↾ 𝑀) “ (◡(𝐹 ↾ 𝑊) “ ((𝐹 ↾ 𝑊) “ 𝑥))) = (◡(𝐻 ↾ 𝑀) “ 𝑥))
40		imaco 6245	. . . . . . 7 ⊢ ((◡(𝐻 ↾ 𝑀) ∘ ◡(𝐹 ↾ 𝑊)) “ ((𝐹 ↾ 𝑊) “ 𝑥)) = (◡(𝐻 ↾ 𝑀) “ (◡(𝐹 ↾ 𝑊) “ ((𝐹 ↾ 𝑊) “ 𝑥)))
41		cnvco 5867	. . . . . . . . 9 ⊢ ◡((𝐹 ↾ 𝑊) ∘ (𝐻 ↾ 𝑀)) = (◡(𝐻 ↾ 𝑀) ∘ ◡(𝐹 ↾ 𝑊))
42		cores 6243	. . . . . . . . . . . . 13 ⊢ (ran (𝐻 ↾ 𝑀) ⊆ 𝑊 → ((𝐹 ↾ 𝑊) ∘ (𝐻 ↾ 𝑀)) = (𝐹 ∘ (𝐻 ↾ 𝑀)))
43	13, 42	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹 ↾ 𝑊) ∘ (𝐻 ↾ 𝑀)) = (𝐹 ∘ (𝐻 ↾ 𝑀)))
44		resco 6244	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ 𝐻) ↾ 𝑀) = (𝐹 ∘ (𝐻 ↾ 𝑀))
45	43, 44	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ↾ 𝑊) ∘ (𝐻 ↾ 𝑀)) = ((𝐹 ∘ 𝐻) ↾ 𝑀))
46	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → ((𝐹 ↾ 𝑊) ∘ (𝐻 ↾ 𝑀)) = ((𝐹 ∘ 𝐻) ↾ 𝑀))
47	46	cnveqd 5853	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → ◡((𝐹 ↾ 𝑊) ∘ (𝐻 ↾ 𝑀)) = ◡((𝐹 ∘ 𝐻) ↾ 𝑀))
48	41, 47	eqtr3id 2810	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡(𝐻 ↾ 𝑀) ∘ ◡(𝐹 ↾ 𝑊)) = ◡((𝐹 ∘ 𝐻) ↾ 𝑀))
49	48	imaeq1d 6053	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → ((◡(𝐻 ↾ 𝑀) ∘ ◡(𝐹 ↾ 𝑊)) “ ((𝐹 ↾ 𝑊) “ 𝑥)) = (◡((𝐹 ∘ 𝐻) ↾ 𝑀) “ ((𝐹 ↾ 𝑊) “ 𝑥)))
50	40, 49	eqtr3id 2810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡(𝐻 ↾ 𝑀) “ (◡(𝐹 ↾ 𝑊) “ ((𝐹 ↾ 𝑊) “ 𝑥))) = (◡((𝐹 ∘ 𝐻) ↾ 𝑀) “ ((𝐹 ↾ 𝑊) “ 𝑥)))
51	39, 50	eqtr3d 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡(𝐻 ↾ 𝑀) “ 𝑥) = (◡((𝐹 ∘ 𝐻) ↾ 𝑀) “ ((𝐹 ↾ 𝑊) “ 𝑥)))
52		cvmlift2lem9a.g	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
53		cvmlift2lem9a.y	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐾
54	53	cnrest 23583	. . . . . . . 8 ⊢ (((𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽) ∧ 𝑀 ⊆ 𝑌) → ((𝐹 ∘ 𝐻) ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐽))
55	52, 8, 54	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐽))
56	55	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → ((𝐹 ∘ 𝐻) ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐽))
57		resima2 6007	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑊 → ((𝐹 ↾ 𝑊) “ 𝑥) = (𝐹 “ 𝑥))
58	36, 57	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → ((𝐹 ↾ 𝑊) “ 𝑥) = (𝐹 “ 𝑥))
59	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
60		restopn2 23475	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Top ∧ 𝑊 ∈ 𝐶) → (𝑥 ∈ (𝐶 ↾t 𝑊) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ⊆ 𝑊)))
61	3, 30, 60	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐶 ↾t 𝑊) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ⊆ 𝑊)))
62	61	simprbda 504	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → 𝑥 ∈ 𝐶)
63		cvmopn 36014	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ 𝐶) → (𝐹 “ 𝑥) ∈ 𝐽)
64	59, 62, 63	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (𝐹 “ 𝑥) ∈ 𝐽)
65	58, 64	eqeltrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → ((𝐹 ↾ 𝑊) “ 𝑥) ∈ 𝐽)
66		cnima 23563	. . . . . 6 ⊢ ((((𝐹 ∘ 𝐻) ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐽) ∧ ((𝐹 ↾ 𝑊) “ 𝑥) ∈ 𝐽) → (◡((𝐹 ∘ 𝐻) ↾ 𝑀) “ ((𝐹 ↾ 𝑊) “ 𝑥)) ∈ (𝐾 ↾t 𝑀))
67	56, 65, 66	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡((𝐹 ∘ 𝐻) ↾ 𝑀) “ ((𝐹 ↾ 𝑊) “ 𝑥)) ∈ (𝐾 ↾t 𝑀))
68	51, 67	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶 ↾t 𝑊)) → (◡(𝐻 ↾ 𝑀) “ 𝑥) ∈ (𝐾 ↾t 𝑀))
69	68	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐶 ↾t 𝑊)(◡(𝐻 ↾ 𝑀) “ 𝑥) ∈ (𝐾 ↾t 𝑀))
70		cvmlift2lem9a.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
71	53	toptopon 23215	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
72	70, 71	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
73		resttopon 23459	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀 ⊆ 𝑌) → (𝐾 ↾t 𝑀) ∈ (TopOn‘𝑀))
74	72, 8, 73	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐾 ↾t 𝑀) ∈ (TopOn‘𝑀))
75		iscn 23533	. . . 4 ⊢ (((𝐾 ↾t 𝑀) ∈ (TopOn‘𝑀) ∧ (𝐶 ↾t 𝑊) ∈ (TopOn‘𝑊)) → ((𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn (𝐶 ↾t 𝑊)) ↔ ((𝐻 ↾ 𝑀):𝑀⟶𝑊 ∧ ∀𝑥 ∈ (𝐶 ↾t 𝑊)(◡(𝐻 ↾ 𝑀) “ 𝑥) ∈ (𝐾 ↾t 𝑀))))
76	74, 34, 75	syl2anc 596	. . 3 ⊢ (𝜑 → ((𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn (𝐶 ↾t 𝑊)) ↔ ((𝐻 ↾ 𝑀):𝑀⟶𝑊 ∧ ∀𝑥 ∈ (𝐶 ↾t 𝑊)(◡(𝐻 ↾ 𝑀) “ 𝑥) ∈ (𝐾 ↾t 𝑀))))
77	15, 69, 76	mpbir2and 726	. 2 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn (𝐶 ↾t 𝑊)))
78	5, 77	sseldd 3932	1 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412   ↾_t crest 17571  Topctop 23191  TopOnctopon 23208   Cn ccn 23522  Homeochmeo 24052   CovMap ccvm 35989

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-map 8833  df-en 8958  df-fin 8961  df-fi 9387  df-rest 17573  df-topgen 17594  df-top 23192  df-topon 23209  df-bases 23244  df-cn 23525  df-hmeo 24054  df-cvm 35990

This theorem is used by:  cvmlift2lem9  36045  cvmlift3lem7  36059