Theorem cvmlift3lem7 35293

Description: Lemma for cvmlift3 35296. (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 35289	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
13	1, 2, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem5 35291	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
14	13, 8	eqeltrd 2844	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))
15		sconntop 35196	. . . . 5 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		cvmlift3lem7.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))
18		cnvimass 6111	. . . . . . 7 ⊢ (◡𝐺 “ 𝐴) ⊆ dom 𝐺
19		eqid 2740	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
20	2, 19	cnf 23275	. . . . . . . 8 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
21		fdm 6756	. . . . . . . 8 ⊢ (𝐺:𝑌⟶∪ 𝐽 → dom 𝐺 = 𝑌)
22	8, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝑌)
23	18, 22	sseqtrid 4061	. . . . . 6 ⊢ (𝜑 → (◡𝐺 “ 𝐴) ⊆ 𝑌)
24	17, 23	sstrd 4019	. . . . 5 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
25		cvmlift3lem7.5	. . . . . 6 ⊢ (𝜑 → 𝑉 ⊆ 𝑀)
26		cvmlift3lem7.6	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
27	25, 26	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
28	24, 27	sseldd 4009	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
29		cvmlift3lem7.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
30	12, 28	ffvelcdmd 7119	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
31		fvco3 7021	. . . . . . . 8 ⊢ ((𝐻:𝑌⟶𝐵 ∧ 𝑋 ∈ 𝑌) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
32	12, 28, 31	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
33	13	fveq1d 6922	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
34	32, 33	eqtr3d 2782	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
35		cvmlift3lem7.1	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
36	34, 35	eqeltrd 2844	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
37		cvmlift3lem7.w	. . . . . 6 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
38	3, 1, 37	cvmsiota 35245	. . . . 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 1372	. . . 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 35290	. . . . . . . . . . 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 686	. . . . . . . . 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 6919	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
46	45	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
47		fveq1 6919	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
48	47	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
49		coeq2 5883	. . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
52	51	riotabidv 7406	. . . . . . . . . . . . 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 5883	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝐹 ∘ 𝑎) = (𝐹 ∘ 𝑔))
54	53	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
55		fveq1 6919	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
56	55	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
57	54, 56	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ ℎ) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
58	57	cbvriotavw 7414	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 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 6922	. . . . . . . . . . 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 1436	. . . . . . . . 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 3244	. . . . . . . 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 23191	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ 𝑀 ⊆ 𝑌) → 𝑀 = ∪ (𝐾 ↾t 𝑀))
68	16, 24, 67	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 = ∪ (𝐾 ↾t 𝑀))
69	27, 68	eleqtrd 2846	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
70	69	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑋 ∈ ∪ (𝐾 ↾t 𝑀))
71	68	eleq2d 2830	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝑀 ↔ 𝑦 ∈ ∪ (𝐾 ↾t 𝑀)))
72	71	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ∪ (𝐾 ↾t 𝑀))
73		eqid 2740	. . . . . . . . 9 ⊢ ∪ (𝐾 ↾t 𝑀) = ∪ (𝐾 ↾t 𝑀)
74	73	pconncn 35192	. . . . . . . 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 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∃𝑛 ∈ (II Cn (𝐾 ↾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76		reeanv 3235	. . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 729	. . . . . . . . . . 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 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 631	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑔 → (((𝐹 ∘ 𝑎) = (𝐺 ∘ 𝑛) ∧ (𝑎‘0) = (𝐻‘𝑋)) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑛) ∧ (𝑔‘0) = (𝐻‘𝑋))))
96	95	cbvriotavw 7414	. . . . . . . . . . 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 35292	. . . . . . . . . 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 3219	. . . . . . . 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 698	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝐻‘𝑦) ∈ 𝑊)
102	101	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊)
103	12	ffund 6751	. . . . . 6 ⊢ (𝜑 → Fun 𝐻)
104	12	fdmd 6757	. . . . . . 7 ⊢ (𝜑 → dom 𝐻 = 𝑌)
105	24, 104	sseqtrrd 4050	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ dom 𝐻)
106		funimass4 6986	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑀 ⊆ dom 𝐻) → ((𝐻 “ 𝑀) ⊆ 𝑊 ↔ ∀𝑦 ∈ 𝑀 (𝐻‘𝑦) ∈ 𝑊))
107	103, 105, 106	syl2anc 583	. . . . 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 35271	. . 3 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
110	73	cncnpi 23307	. . 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 583	. 2 ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ (((𝐾 ↾t 𝑀) CnP 𝐶)‘𝑋))
112		cvmlift3lem7.4	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐾)
113	2	ssntr 23087	. . . . 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 4009	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((int‘𝐾)‘𝑀))
116	2, 1	cnprest 23318	. . 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249  ^◡ccnv 5699  dom cdm 5700   ↾ cres 5702   “ cima 5703   ∘ ccom 5704  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  0cc0 11184  1c1 11185   ↾_t crest 17480  Topctop 22920  intcnt 23046   Cn ccn 23253   CnP ccnp 23254  𝑛-Locally cnlly 23494  Homeochmeo 23782  IIcii 24920  PConncpconn 35187  SConncsconn 35188   CovMap ccvm 35223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-ec 8765  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-rest 17482  df-topn 17483  df-0g 17501  df-gsum 17502  df-topgen 17503  df-pt 17504  df-prds 17507  df-xrs 17562  df-qtop 17567  df-imas 17568  df-xps 17570  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-mulg 19108  df-cntz 19357  df-cmn 19824  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-cnfld 21388  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-cn 23256  df-cnp 23257  df-cmp 23416  df-conn 23441  df-lly 23495  df-nlly 23496  df-tx 23591  df-hmeo 23784  df-xms 24351  df-ms 24352  df-tms 24353  df-ii 24922  df-cncf 24923  df-htpy 25021  df-phtpy 25022  df-phtpc 25043  df-pco 25057  df-pconn 35189  df-sconn 35190  df-cvm 35224

This theorem is referenced by:  cvmlift3lem8  35294