Theorem cvmlift3lem7 35369

Description: Lemma for cvmlift3 35372. (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 35365	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
13	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem5 35367	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
14	13, 8	eqeltrd 2831	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15		sconntop 35272	. . . . 5 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		cvmlift3lem7.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))
18		cnvimass 6030	. . . . . . 7 ⊢ (◡𝐺 “ 𝐴) ⊆ dom 𝐺
19		eqid 2731	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
20	2, 19	cnf 23161	. . . . . . . 8 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
21		fdm 6660	. . . . . . . 8 ⊢ (𝐺:𝑌⟶∪ 𝐽 → dom 𝐺 = 𝑌)
22	8, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝑌)
23	18, 22	sseqtrid 3972	. . . . . 6 ⊢ (𝜑 → (◡𝐺 “ 𝐴) ⊆ 𝑌)
24	17, 23	sstrd 3940	. . . . 5 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
25		cvmlift3lem7.5	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑀)
26		cvmlift3lem7.6	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	25, 26	sseldd 3930	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
28	24, 27	sseldd 3930	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
29		cvmlift3lem7.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
30	12, 28	ffvelcdmd 7018	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
31		fvco3 6921	. . . . . . . 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 6824	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
34	32, 33	eqtr3d 2768	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
35		cvmlift3lem7.1	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
36	34, 35	eqeltrd 2831	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
37		cvmlift3lem7.w	. . . . . 6 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
38	3, 1, 37	cvmsiota 35321	. . . . 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 2731	. . . . . . . . . . 11 ⊢ (𝐻‘𝑋) = (𝐻‘𝑋)
41	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem4 35366	. . . . . . . . . . 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 6821	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
46	45	eqeq1d 2733	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
47		fveq1 6821	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
48	47	eqeq1d 2733	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
49		coeq2 5797	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
50	49	eqeq2d 2742	. . . . . . . . . . . . . . 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 7305	. . . . . . . . . . . . 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 5797	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝐹 ∘ 𝑎) = (𝐹 ∘ 𝑔))
54	53	eqeq1d 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
55		fveq1 6821	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
56	55	eqeq1d 2733	. . . . . . . . . . . . . . 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 7313	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 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 6824	. . . . . . . . . . 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 2733	. . . . . . . . . 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 3211	. . . . . . . 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 23077	. . . . . . . . . . 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 2833	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
70	69	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
71	68	eleq2d 2817	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ ∪ (𝐾 ↾t 𝑀)))
72	71	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ∪ (𝐾 ↾t 𝑀))
73		eqid 2731	. . . . . . . . 9 ⊢ ∪ (𝐾 ↾t 𝑀) = ∪ (𝐾 ↾t 𝑀)
74	73	pconncn 35268	. . . . . . . 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 3204	. . . . . . . 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 2733	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛)))
94	55	eqeq1d 2733	. . . . . . . . . . . . 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 7313	. . . . . . . . . . 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 35368	. . . . . . . . . 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 3189	. . . . . . . 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 3124	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊)
103	12	ffund 6655	. . . . . 6 ⊢ (𝜑 → Fun 𝐻)
104	12	fdmd 6661	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = 𝑌)
105	24, 104	sseqtrrd 3967	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ dom 𝐻)
106		funimass4 6886	. . . . . 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 35347	. . 3 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
110	73	cncnpi 23193	. . 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 22973	. . . . 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 3930	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((int‘𝐾)‘𝑀))
116	2, 1	cnprest 23204	. . 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 2111  ∀wral 3047  ∃wrex 3056  {crab 3395   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573  ∪ cuni 4856   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614   ↾ cres 5616   “ cima 5617   ∘ ccom 5618  Fun wfun 6475  ⟶wf 6477  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  0cc0 11006  1c1 11007   ↾_t crest 17324  Topctop 22808  intcnt 22932   Cn ccn 23139   CnP ccnp 23140  𝑛-Locally cnlly 23380  Homeochmeo 23668  IIcii 24795  PConncpconn 35263  SConncsconn 35264   CovMap ccvm 35299

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084  ax-addf 11085

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-ec 8624  df-map 8752  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-fi 9295  df-sup 9326  df-inf 9327  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-q 12847  df-rp 12891  df-xneg 13011  df-xadd 13012  df-xmul 13013  df-ioo 13249  df-ico 13251  df-icc 13252  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-rest 17326  df-topn 17327  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-prds 17351  df-xrs 17406  df-qtop 17411  df-imas 17412  df-xps 17414  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-submnd 18692  df-mulg 18981  df-cntz 19229  df-cmn 19694  df-psmet 21283  df-xmet 21284  df-met 21285  df-bl 21286  df-mopn 21287  df-cnfld 21292  df-top 22809  df-topon 22826  df-topsp 22848  df-bases 22861  df-cld 22934  df-ntr 22935  df-cls 22936  df-nei 23013  df-cn 23142  df-cnp 23143  df-cmp 23302  df-conn 23327  df-lly 23381  df-nlly 23382  df-tx 23477  df-hmeo 23670  df-xms 24235  df-ms 24236  df-tms 24237  df-ii 24797  df-cncf 24798  df-htpy 24896  df-phtpy 24897  df-phtpc 24918  df-pco 24932  df-pconn 35265  df-sconn 35266  df-cvm 35300

This theorem is referenced by:  cvmlift3lem8  35370