Theorem cvmlift3lem7 32187

Description: Lemma for cvmlift3 32190. (Contributed by Mario Carneiro, 9-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
cvmlift3.h	⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
cvmlift3lem7.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))))})
cvmlift3lem7.1	⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
cvmlift3lem7.2	⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
cvmlift3lem7.3	⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))
cvmlift3lem7.w	⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
cvmlift3lem7.7	⊢ (𝜑 → (𝐾 ↾t 𝑀) ∈ PConn)
cvmlift3lem7.4	⊢ (𝜑 → 𝑉 ∈ 𝐾)
cvmlift3lem7.5	⊢ (𝜑 → 𝑉 ⊆ 𝑀)
cvmlift3lem7.6	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
cvmlift3lem7	⊢ (𝜑 → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))

Distinct variable groups:   𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝐴   𝑓,𝑔,𝑧,𝑏,𝑥   𝐽,𝑏   𝑔,𝑐,𝑥,𝐽,𝑑,𝑓,𝑘,𝑠   𝐹,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠   𝑥,𝑧,𝐹   𝑓,𝑀,𝑔,𝑥   𝐻,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑆,𝑏,𝑓,𝑥   𝐵,𝑏,𝑑,𝑓,𝑔,𝑥,𝑧   𝑋,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝐺,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧   𝑇,𝑏,𝑐,𝑑,𝑠   𝐶,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠,𝑥,𝑧   𝜑,𝑓,𝑥   𝐾,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧   𝑃,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑂,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧   𝑊,𝑐,𝑑,𝑓,𝑥

Allowed substitution hints:   𝜑(𝑧,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑)   𝐴(𝑥,𝑔)   𝐵(𝑘,𝑠,𝑐)   𝑃(𝑘,𝑠)   𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝑇(𝑥,𝑧,𝑓,𝑔,𝑘)   𝐺(𝑠)   𝐻(𝑘,𝑠)   𝐽(𝑧)   𝐾(𝑘,𝑠,𝑑)   𝑀(𝑧,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑂(𝑘,𝑠,𝑑)   𝑉(𝑥,𝑧,𝑓,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑊(𝑧,𝑔,𝑘,𝑠,𝑏)   𝑋(𝑘,𝑠)   𝑌(𝑘,𝑠,𝑏,𝑐,𝑑)

Proof of Theorem cvmlift3lem7

Dummy variables 𝑎 𝑦 ℎ 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift3.b	. . . 4 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift3.y	. . . 4 ⊢ 𝑌 = ∪ 𝐾
3		cvmlift3lem7.s	. . . 4 ⊢ 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))))})
4		cvmlift3.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		cvmlift3.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ SConn)
6		cvmlift3.l	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
7		cvmlift3.o	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
8		cvmlift3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
9		cvmlift3.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
10		cvmlift3.e	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
11		cvmlift3.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
12	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem3 32183	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
13	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem5 32185	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
14	13, 8	eqeltrd 2883	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15		sconntop 32090	. . . . 5 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		cvmlift3lem7.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))
18		cnvimass 5830	. . . . . . 7 ⊢ (◡𝐺 “ 𝐴) ⊆ dom 𝐺
19		eqid 2795	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
20	2, 19	cnf 21543	. . . . . . . 8 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
21		fdm 6395	. . . . . . . 8 ⊢ (𝐺:𝑌⟶∪ 𝐽 → dom 𝐺 = 𝑌)
22	8, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝑌)
23	18, 22	sseqtrid 3944	. . . . . 6 ⊢ (𝜑 → (◡𝐺 “ 𝐴) ⊆ 𝑌)
24	17, 23	sstrd 3903	. . . . 5 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
25		cvmlift3lem7.5	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑀)
26		cvmlift3lem7.6	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	25, 26	sseldd 3894	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
28	24, 27	sseldd 3894	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
29		cvmlift3lem7.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
30	12, 28	ffvelrnd 6722	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
31		fvco3 6632	. . . . . . . 8 ⊢ ((𝐻:𝑌⟶𝐵 ∧ 𝑋 ∈ 𝑌) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
32	12, 28, 31	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
33	13	fveq1d 6545	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
34	32, 33	eqtr3d 2833	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
35		cvmlift3lem7.1	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
36	34, 35	eqeltrd 2883	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
37		cvmlift3lem7.w	. . . . . 6 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
38	3, 1, 37	cvmsiota 32139	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝐴) ∧ (𝐻‘𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)) → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
39	4, 29, 30, 36, 38	syl13anc 1365	. . . 4 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
40		eqid 2795	. . . . . . . . . . 11 ⊢ (𝐻‘𝑋) = (𝐻‘𝑋)
41	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem4 32184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = (𝐻‘𝑋) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋))))
42	40, 41	mpbii 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
43	28, 42	mpdan 683	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
44	43	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
45		fveq1 6542	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
46	45	eqeq1d 2797	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
47		fveq1 6542	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
48	47	eqeq1d 2797	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
49		coeq2 5620	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
50	49	eqeq2d 2805	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
51	50	anbi1d 629	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
52	51	riotabidv 6984	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
53		coeq2 5620	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝐹 ∘ 𝑎) = (𝐹 ∘ 𝑔))
54	53	eqeq1d 2797	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
55		fveq1 6542	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
56	55	eqeq1d 2797	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
57	54, 56	anbi12d 630	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
58	57	cbvriotav 6993	. . . . . . . . . . . . 13 ⊢ (℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))
59	52, 58	syl6eqr 2849	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃)))
60	59	fveq1d 6545	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1))
61	60	eqeq1d 2797	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋) ↔ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
62	46, 48, 61	3anbi123d 1428	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ↔ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋))))
63	62	cbvrexv 3404	. . . . . . . 8 ⊢ (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ↔ ∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
64	44, 63	sylib 219	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
65		cvmlift3lem7.7	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾t 𝑀) ∈ PConn)
66	65	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝐾 ↾t 𝑀) ∈ PConn)
67	2	restuni 21459	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) → 𝑀 = ∪ (𝐾 ↾t 𝑀))
68	16, 24, 67	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 = ∪ (𝐾 ↾t 𝑀))
69	27, 68	eleqtrd 2885	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
70	69	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
71	68	eleq2d 2868	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ ∪ (𝐾 ↾t 𝑀)))
72	71	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ∪ (𝐾 ↾t 𝑀))
73		eqid 2795	. . . . . . . . 9 ⊢ ∪ (𝐾 ↾t 𝑀) = ∪ (𝐾 ↾t 𝑀)
74	73	pconncn 32086	. . . . . . . 8 ⊢ (((𝐾 ↾t 𝑀) ∈ PConn ∧ 𝑋 ∈ ∪ (𝐾 ↾t 𝑀) ∧ 𝑦 ∈ ∪ (𝐾 ↾t 𝑀)) → ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
75	66, 70, 72, 74	syl3anc 1364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76		reeanv 3328	. . . . . . . 8 ⊢ (∃ℎ ∈ (II Cn 𝐾)∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))(((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) ↔ (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)))
77	4	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
78	5	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐾 ∈ SConn)
79	6	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐾 ∈ 𝑛-Locally PConn)
80	7	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑂 ∈ 𝑌)
81	8	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐺 ∈ (𝐾 Cn 𝐽))
82	9	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑃 ∈ 𝐵)
83	10	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐹‘𝑃) = (𝐺‘𝑂))
84	35	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐺‘𝑋) ∈ 𝐴)
85	29	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑇 ∈ (𝑆‘𝐴))
86	17	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑀 ⊆ (◡𝐺 “ 𝐴))
87	27	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑋 ∈ 𝑀)
88		simpllr 772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑦 ∈ 𝑀)
89		simplrl 773	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ℎ ∈ (II Cn 𝐾))
90		simprl 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
91		simplrr 774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))
92		simprr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
93	53	eqeq1d 2797	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
94	55	eqeq1d 2797	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻‘𝑋) ↔ (𝑔‘0) = (𝐻‘𝑋)))
95	93, 94	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ∧ (𝑎‘0) = (𝐻‘𝑋)) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (𝑔‘0) = (𝐻‘𝑋))))
96	95	cbvriotav 6993	. . . . . . . . . . 11 ⊢ (℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ∧ (𝑎‘0) = (𝐻‘𝑋))) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (𝑔‘0) = (𝐻‘𝑋)))
97	1, 2, 77, 78, 79, 80, 81, 82, 83, 11, 3, 84, 85, 86, 37, 87, 88, 89, 58, 90, 91, 92, 96	cvmlift3lem6 32186	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) ∧ (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐻‘𝑦) ∈ 𝑊)
98	97	ex 413	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ (ℎ ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾 ↾t 𝑀)))) → ((((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻‘𝑦) ∈ 𝑊))
99	98	rexlimdvva 3257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (∃ℎ ∈ (II Cn 𝐾)∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))(((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻‘𝑦) ∈ 𝑊))
100	76, 99	syl5bir 244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)) ∧ ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻‘𝑦) ∈ 𝑊))
101	64, 75, 100	mp2and 695	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝐻‘𝑦) ∈ 𝑊)
102	101	ralrimiva 3149	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊)
103	12	ffund 6391	. . . . . 6 ⊢ (𝜑 → Fun 𝐻)
104	12	fdmd 6396	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = 𝑌)
105	24, 104	sseqtr4d 3933	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ dom 𝐻)
106		funimass4 6603	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑀 ⊆ dom 𝐻) → ((𝐻 “ 𝑀) ⊆ 𝑊 ↔ ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊))
107	103, 105, 106	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝐻 “ 𝑀) ⊆ 𝑊 ↔ ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊))
108	102, 107	mpbird 258	. . . 4 ⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊)
109	1, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 108	cvmlift2lem9a 32165	. . 3 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
110	73	cncnpi 21575	. . 3 ⊢ (((𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶) ∧ 𝑋 ∈ ∪ (𝐾 ↾t 𝑀)) → (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋))
111	109, 69, 110	syl2anc 584	. 2 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋))
112		cvmlift3lem7.4	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐾)
113	2	ssntr 21355	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) ∧ (𝑉 ∈ 𝐾 ∧ 𝑉 ⊆ 𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
114	16, 24, 112, 25, 113	syl22anc 835	. . . 4 ⊢ (𝜑 → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
115	114, 26	sseldd 3894	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((int‘𝐾)‘𝑀))
116	2, 1	cnprest 21586	. . 3 ⊢ (((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) ∧ (𝑋 ∈ ((int‘𝐾)‘𝑀) ∧ 𝐻:𝑌⟶𝐵)) → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋)))
117	16, 24, 115, 12, 116	syl22anc 835	. 2 ⊢ (𝜑 → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋)))
118	111, 117	mpbird 258	1 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  {crab 3109   ∖ cdif 3860   ∩ cin 3862   ⊆ wss 3863  ∅c0 4215  𝒫 cpw 4457  {csn 4476  ∪ cuni 4749   ↦ cmpt 5045  ^◡ccnv 5447  dom cdm 5448   ↾ cres 5450   “ cima 5451   ∘ ccom 5452  Fun wfun 6224  ⟶wf 6226  ‘cfv 6230  ℩crio 6981  (class class class)co 7021  0cc0 10388  1c1 10389   ↾_t crest 16528  Topctop 21190  intcnt 21314   Cn ccn 21521   CnP ccnp 21522  𝑛-Locally cnlly 21762  Homeochmeo 22050  IIcii 23171  PConncpconn 32081  SConncsconn 32082   CovMap ccvm 32117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-inf2 8955  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466  ax-addf 10467  ax-mulf 10468

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-iin 4832  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-se 5408  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-isom 6239  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-of 7272  df-om 7442  df-1st 7550  df-2nd 7551  df-supp 7687  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-2o 7959  df-oadd 7962  df-er 8144  df-ec 8146  df-map 8263  df-ixp 8316  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-fsupp 8685  df-fi 8726  df-sup 8757  df-inf 8758  df-oi 8825  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-2 11553  df-3 11554  df-4 11555  df-5 11556  df-6 11557  df-7 11558  df-8 11559  df-9 11560  df-n0 11751  df-z 11835  df-dec 11953  df-uz 12099  df-q 12203  df-rp 12245  df-xneg 12362  df-xadd 12363  df-xmul 12364  df-ioo 12597  df-ico 12599  df-icc 12600  df-fz 12748  df-fzo 12889  df-fl 13017  df-seq 13225  df-exp 13285  df-hash 13546  df-cj 14297  df-re 14298  df-im 14299  df-sqrt 14433  df-abs 14434  df-clim 14684  df-sum 14882  df-struct 16319  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-mulr 16413  df-starv 16414  df-sca 16415  df-vsca 16416  df-ip 16417  df-tset 16418  df-ple 16419  df-ds 16421  df-unif 16422  df-hom 16423  df-cco 16424  df-rest 16530  df-topn 16531  df-0g 16549  df-gsum 16550  df-topgen 16551  df-pt 16552  df-prds 16555  df-xrs 16609  df-qtop 16614  df-imas 16615  df-xps 16617  df-mre 16691  df-mrc 16692  df-acs 16694  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-submnd 17780  df-mulg 17987  df-cntz 18193  df-cmn 18640  df-psmet 20224  df-xmet 20225  df-met 20226  df-bl 20227  df-mopn 20228  df-cnfld 20233  df-top 21191  df-topon 21208  df-topsp 21230  df-bases 21243  df-cld 21316  df-ntr 21317  df-cls 21318  df-nei 21395  df-cn 21524  df-cnp 21525  df-cmp 21684  df-conn 21709  df-lly 21763  df-nlly 21764  df-tx 21859  df-hmeo 22052  df-xms 22618  df-ms 22619  df-tms 22620  df-ii 23173  df-htpy 23262  df-phtpy 23263  df-phtpc 23284  df-pco 23297  df-pconn 32083  df-sconn 32084  df-cvm 32118

This theorem is referenced by:  cvmlift3lem8  32188