Theorem cvmlift3lem7 35390

Description: Lemma for cvmlift3 35393. (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 35386	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
13	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem5 35388	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
14	13, 8	eqeltrd 2833	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15		sconntop 35293	. . . . 5 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		cvmlift3lem7.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))
18		cnvimass 6035	. . . . . . 7 ⊢ (◡𝐺 “ 𝐴) ⊆ dom 𝐺
19		eqid 2733	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
20	2, 19	cnf 23162	. . . . . . . 8 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
21		fdm 6665	. . . . . . . 8 ⊢ (𝐺:𝑌⟶∪ 𝐽 → dom 𝐺 = 𝑌)
22	8, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝑌)
23	18, 22	sseqtrid 3973	. . . . . 6 ⊢ (𝜑 → (◡𝐺 “ 𝐴) ⊆ 𝑌)
24	17, 23	sstrd 3941	. . . . 5 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
25		cvmlift3lem7.5	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑀)
26		cvmlift3lem7.6	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	25, 26	sseldd 3931	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
28	24, 27	sseldd 3931	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
29		cvmlift3lem7.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
30	12, 28	ffvelcdmd 7024	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
31		fvco3 6927	. . . . . . . 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 6830	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
34	32, 33	eqtr3d 2770	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
35		cvmlift3lem7.1	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
36	34, 35	eqeltrd 2833	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
37		cvmlift3lem7.w	. . . . . 6 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
38	3, 1, 37	cvmsiota 35342	. . . . 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 1374	. . . 4 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
40		eqid 2733	. . . . . . . . . . 11 ⊢ (𝐻‘𝑋) = (𝐻‘𝑋)
41	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem4 35387	. . . . . . . . . . 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 233	. . . . . . . . . 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 687	. . . . . . . . 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 480	. . . . . . . 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 6827	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
46	45	eqeq1d 2735	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
47		fveq1 6827	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
48	47	eqeq1d 2735	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
49		coeq2 5802	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
50	49	eqeq2d 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
51	50	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
52	51	riotabidv 7311	. . . . . . . . . . . . 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 5802	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝐹 ∘ 𝑎) = (𝐹 ∘ 𝑔))
54	53	eqeq1d 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
55		fveq1 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
56	55	eqeq1d 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
57	54, 56	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
58	57	cbvriotavw 7319	. . . . . . . . . . . . 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	eqtr4di 2786	. . . . . . . . . . . 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 6830	. . . . . . . . . . 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 2735	. . . . . . . . . 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 1438	. . . . . . . . 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	cbvrexvw 3212	. . . . . . . 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 218	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝐾 ↾t 𝑀) ∈ PConn)
67	2	restuni 23078	. . . . . . . . . . 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 2835	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
70	69	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
71	68	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ ∪ (𝐾 ↾t 𝑀)))
72	71	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ∪ (𝐾 ↾t 𝑀))
73		eqid 2733	. . . . . . . . 9 ⊢ ∪ (𝐾 ↾t 𝑀) = ∪ (𝐾 ↾t 𝑀)
74	73	pconncn 35289	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76		reeanv 3205	. . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 775	. . . . . . . . . . 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 776	. . . . . . . . . . 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 770	. . . . . . . . . . 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 777	. . . . . . . . . . 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 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))) → ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
93	53	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
94	55	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻‘𝑋) ↔ (𝑔‘0) = (𝐻‘𝑋)))
95	93, 94	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ∧ (𝑎‘0) = (𝐻‘𝑋)) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (𝑔‘0) = (𝐻‘𝑋))))
96	95	cbvriotavw 7319	. . . . . . . . . . 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 35389	. . . . . . . . . 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 412	. . . . . . . . 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 3190	. . . . . . . 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	biimtrrid 243	. . . . . . 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 699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝐻‘𝑦) ∈ 𝑊)
102	101	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊)
103	12	ffund 6660	. . . . . 6 ⊢ (𝜑 → Fun 𝐻)
104	12	fdmd 6666	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = 𝑌)
105	24, 104	sseqtrrd 3968	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ dom 𝐻)
106		funimass4 6892	. . . . . 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 257	. . . 4 ⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊)
109	1, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 108	cvmlift2lem9a 35368	. . 3 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
110	73	cncnpi 23194	. . 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 22974	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) ∧ (𝑉 ∈ 𝐾 ∧ 𝑉 ⊆ 𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
114	16, 24, 112, 25, 113	syl22anc 838	. . . 4 ⊢ (𝜑 → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
115	114, 26	sseldd 3931	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((int‘𝐾)‘𝑀))
116	2, 1	cnprest 23205	. . 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 838	. 2 ⊢ (𝜑 → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋)))
118	111, 117	mpbird 257	1 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  {crab 3396   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   “ cima 5622   ∘ ccom 5623  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  ℩crio 7308  (class class class)co 7352  0cc0 11013  1c1 11014   ↾_t crest 17326  Topctop 22809  intcnt 22933   Cn ccn 23140   CnP ccnp 23141  𝑛-Locally cnlly 23381  Homeochmeo 23669  IIcii 24796  PConncpconn 35284  SConncsconn 35285   CovMap ccvm 35320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091  ax-addf 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-ec 8630  df-map 8758  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-fi 9302  df-sup 9333  df-inf 9334  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-q 12849  df-rp 12893  df-xneg 13013  df-xadd 13014  df-xmul 13015  df-ioo 13251  df-ico 13253  df-icc 13254  df-fz 13410  df-fzo 13557  df-fl 13698  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-sum 15596  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-starv 17178  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-unif 17186  df-hom 17187  df-cco 17188  df-rest 17328  df-topn 17329  df-0g 17347  df-gsum 17348  df-topgen 17349  df-pt 17350  df-prds 17353  df-xrs 17408  df-qtop 17413  df-imas 17414  df-xps 17416  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-mulg 18983  df-cntz 19231  df-cmn 19696  df-psmet 21285  df-xmet 21286  df-met 21287  df-bl 21288  df-mopn 21289  df-cnfld 21294  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cld 22935  df-ntr 22936  df-cls 22937  df-nei 23014  df-cn 23143  df-cnp 23144  df-cmp 23303  df-conn 23328  df-lly 23382  df-nlly 23383  df-tx 23478  df-hmeo 23671  df-xms 24236  df-ms 24237  df-tms 24238  df-ii 24798  df-cncf 24799  df-htpy 24897  df-phtpy 24898  df-phtpc 24919  df-pco 24933  df-pconn 35286  df-sconn 35287  df-cvm 35321

This theorem is referenced by:  cvmlift3lem8  35391