Theorem cvmlift3lem7 35560

Description: Lemma for cvmlift3 35563. (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 35556	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
13	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem5 35558	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
14	13, 8	eqeltrd 2840	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15		sconntop 35463	. . . . 5 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		cvmlift3lem7.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))
18		cnvimass 6041	. . . . . . 7 ⊢ (◡𝐺 “ 𝐴) ⊆ dom 𝐺
19		eqid 2740	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
20	2, 19	cnf 23236	. . . . . . . 8 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
21		fdm 6671	. . . . . . . 8 ⊢ (𝐺:𝑌⟶∪ 𝐽 → dom 𝐺 = 𝑌)
22	8, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝑌)
23	18, 22	sseqtrid 3964	. . . . . 6 ⊢ (𝜑 → (◡𝐺 “ 𝐴) ⊆ 𝑌)
24	17, 23	sstrd 3932	. . . . 5 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
25		cvmlift3lem7.5	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑀)
26		cvmlift3lem7.6	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	25, 26	sseldd 3923	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
28	24, 27	sseldd 3923	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
29		cvmlift3lem7.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
30	12, 28	ffvelcdmd 7033	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
31		fvco3 6934	. . . . . . . 8 ⊢ ((𝐻:𝑌⟶𝐵 ∧ 𝑋 ∈ 𝑌) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
32	12, 28, 31	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
33	13	fveq1d 6836	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
34	32, 33	eqtr3d 2777	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
35		cvmlift3lem7.1	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
36	34, 35	eqeltrd 2840	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
37		cvmlift3lem7.w	. . . . . 6 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
38	3, 1, 37	cvmsiota 35512	. . . . 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 1380	. . . 4 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
40		eqid 2740	. . . . . . . . . . 11 ⊢ (𝐻‘𝑋) = (𝐻‘𝑋)
41	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem4 35557	. . . . . . . . . . 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 693	. . . . . . . . 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 6833	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
46	45	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
47		fveq1 6833	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
48	47	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
49		coeq2 5807	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
50	49	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
51	50	anbi1d 637	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
52	51	riotabidv 7322	. . . . . . . . . . . . 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 5807	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝐹 ∘ 𝑎) = (𝐹 ∘ 𝑔))
54	53	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
55		fveq1 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
56	55	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
57	54, 56	anbi12d 638	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
58	57	cbvriotavw 7330	. . . . . . . . . . . . 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 2793	. . . . . . . . . . . 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 6836	. . . . . . . . . . 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 2742	. . . . . . . . . 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 1444	. . . . . . . . 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 3219	. . . . . . . 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 23152	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) → 𝑀 = ∪ (𝐾 ↾t 𝑀))
68	16, 24, 67	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 = ∪ (𝐾 ↾t 𝑀))
69	27, 68	eleqtrd 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
70	69	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
71	68	eleq2d 2826	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ ∪ (𝐾 ↾t 𝑀)))
72	71	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ∪ (𝐾 ↾t 𝑀))
73		eqid 2740	. . . . . . . . 9 ⊢ ∪ (𝐾 ↾t 𝑀) = ∪ (𝐾 ↾t 𝑀)
74	73	pconncn 35459	. . . . . . . 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 1379	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76		reeanv 3212	. . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 781	. . . . . . . . . . 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 782	. . . . . . . . . . 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 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‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑎 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻‘𝑋)))
91		simplrr 783	. . . . . . . . . . 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 778	. . . . . . . . . . 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 2742	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
94	55	eqeq1d 2742	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻‘𝑋) ↔ (𝑔‘0) = (𝐻‘𝑋)))
95	93, 94	anbi12d 638	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ∧ (𝑎‘0) = (𝐻‘𝑋)) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (𝑔‘0) = (𝐻‘𝑋))))
96	95	cbvriotavw 7330	. . . . . . . . . . 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 35559	. . . . . . . . . 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 3197	. . . . . . . 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 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 705	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝐻‘𝑦) ∈ 𝑊)
102	101	ralrimiva 3132	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊)
103	12	ffund 6666	. . . . . 6 ⊢ (𝜑 → Fun 𝐻)
104	12	fdmd 6672	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = 𝑌)
105	24, 104	sseqtrrd 3959	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ dom 𝐻)
106		funimass4 6898	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑀 ⊆ dom 𝐻) → ((𝐻 “ 𝑀) ⊆ 𝑊 ↔ ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊))
107	103, 105, 106	syl2anc 590	. . . . 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 35538	. . 3 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
110	73	cncnpi 23268	. . 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 590	. 2 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋))
112		cvmlift3lem7.4	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐾)
113	2	ssntr 23048	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) ∧ (𝑉 ∈ 𝐾 ∧ 𝑉 ⊆ 𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
114	16, 24, 112, 25, 113	syl22anc 844	. . . 4 ⊢ (𝜑 → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
115	114, 26	sseldd 3923	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((int‘𝐾)‘𝑀))
116	2, 1	cnprest 23279	. . 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 844	. 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 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  {crab 3392   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  {csn 4562  ∪ cuni 4845   ↦ cmpt 5160  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  ℩crio 7319  (class class class)co 7363  0cc0 11036  1c1 11037   ↾_t crest 17381  Topctop 22883  intcnt 23007   Cn ccn 23214   CnP ccnp 23215  𝑛-Locally cnlly 23455  Homeochmeo 23743  IIcii 24867  PConncpconn 35454  SConncsconn 35455   CovMap ccvm 35490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114  ax-addf 11115

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  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-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-ec 8642  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-fi 9321  df-sup 9352  df-inf 9353  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-q 12897  df-rp 12941  df-xneg 13061  df-xadd 13062  df-xmul 13063  df-ioo 13300  df-ico 13302  df-icc 13303  df-fz 13460  df-fzo 13607  df-fl 13749  df-seq 13962  df-exp 14022  df-hash 14291  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-clim 15448  df-sum 15647  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-starv 17233  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-unif 17241  df-hom 17242  df-cco 17243  df-rest 17383  df-topn 17384  df-0g 17402  df-gsum 17403  df-topgen 17404  df-pt 17405  df-prds 17408  df-xrs 17464  df-qtop 17469  df-imas 17470  df-xps 17472  df-mre 17546  df-mrc 17547  df-acs 17549  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-submnd 18750  df-mulg 19042  df-cntz 19290  df-cmn 19755  df-psmet 21346  df-xmet 21347  df-met 21348  df-bl 21349  df-mopn 21350  df-cnfld 21355  df-top 22884  df-topon 22901  df-topsp 22923  df-bases 22936  df-cld 23009  df-ntr 23010  df-cls 23011  df-nei 23088  df-cn 23217  df-cnp 23218  df-cmp 23377  df-conn 23402  df-lly 23456  df-nlly 23457  df-tx 23552  df-hmeo 23745  df-xms 24310  df-ms 24311  df-tms 24312  df-ii 24869  df-cncf 24870  df-htpy 24962  df-phtpy 24963  df-phtpc 24984  df-pco 24997  df-pconn 35456  df-sconn 35457  df-cvm 35491

This theorem is referenced by:  cvmlift3lem8  35561