cvmliftphtlem - Mathbox for Mario Carneiro

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Theorem cvmliftphtlem 35304

Description: Lemma for cvmliftpht 35305. (Contributed by Mario Carneiro, 6-Jul-2015.)

**Hypotheses**
Ref	Expression
cvmliftpht.b	⊢ 𝐵 = ∪ 𝐶
cvmliftpht.m	⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
cvmliftpht.n	⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
cvmliftpht.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftpht.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftpht.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
cvmliftphtlem.g	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
cvmliftphtlem.h	⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
cvmliftphtlem.k	⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))
cvmliftphtlem.a	⊢ (𝜑 → 𝐴 ∈ ((II ×_t II) Cn 𝐶))
cvmliftphtlem.c	⊢ (𝜑 → (𝐹 ∘ 𝐴) = 𝐾)
cvmliftphtlem.0	⊢ (𝜑 → (0𝐴0) = 𝑃)

**Assertion**
Ref	Expression
cvmliftphtlem	⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))

Distinct variable groups: 𝐴,𝑓 𝐵,𝑓 𝑓,𝐹 𝑓,𝐽 𝐶,𝑓 𝑓,𝐺 𝑓,𝐻 𝑃,𝑓 Allowed substitution hints: 𝜑(𝑓) 𝐾(𝑓) 𝑀(𝑓) 𝑁(𝑓)

Proof of Theorem cvmliftphtlem Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftpht.b	. . . 4 ⊢ 𝐵 = ∪ 𝐶
2		cvmliftpht.m	. . . 4 ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
3		cvmliftpht.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmliftphtlem.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
5		cvmliftpht.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
6		cvmliftpht.e	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
7	1, 2, 3, 4, 5, 6	cvmliftiota 35288	. . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
8	7	simp1d 1142	. 2 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐶))
9		cvmliftpht.n	. . . 4 ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
10		cvmliftphtlem.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
11		cvmliftphtlem.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))
12	4, 10, 11	phtpy01 24884	. . . . . 6 ⊢ (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
13	12	simpld 494	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0))
14	6, 13	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
15	1, 9, 3, 10, 5, 14	cvmliftiota 35288	. . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
16	15	simp1d 1142	. 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶))
17		cvmliftphtlem.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ((II ×_t II) Cn 𝐶))
18		iitop 24773	. . . . . . . . . . . . . . . 16 ⊢ II ∈ Top
19		iiuni 24774	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) = ∪ II
20	18, 18, 19, 19	txunii 23480	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
21	20, 1	cnf 23133	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ((II ×_t II) Cn 𝐶) → 𝐴:((0[,]1) × (0[,]1))⟶𝐵)
22	17, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:((0[,]1) × (0[,]1))⟶𝐵)
23		0elunit 13430	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0[,]1)
24		opelxpi 5675	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
25	23, 24	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
26		fvco3 6960	. . . . . . . . . . . . 13 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
27	22, 25, 26	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
28		cvmliftphtlem.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∘ 𝐴) = 𝐾)
29	28	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹 ∘ 𝐴) = 𝐾)
30	29	fveq1d 6860	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 0⟩) = (𝐾‘⟨𝑠, 0⟩))
31	27, 30	eqtr3d 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 0⟩)) = (𝐾‘⟨𝑠, 0⟩))
32		df-ov 7390	. . . . . . . . . . . 12 ⊢ (𝑠𝐴0) = (𝐴‘⟨𝑠, 0⟩)
33	32	fveq2i 6861	. . . . . . . . . . 11 ⊢ (𝐹‘(𝑠𝐴0)) = (𝐹‘(𝐴‘⟨𝑠, 0⟩))
34		df-ov 7390	. . . . . . . . . . 11 ⊢ (𝑠𝐾0) = (𝐾‘⟨𝑠, 0⟩)
35	31, 33, 34	3eqtr4g 2789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝑠𝐾0))
36		iitopon 24772	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
38	4, 10	phtpyhtpy 24881	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ (𝐺(II Htpy 𝐽)𝐻))
39	38, 11	sseldd 3947	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐺(II Htpy 𝐽)𝐻))
40	37, 4, 10, 39	htpyi 24873	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐾0) = (𝐺‘𝑠) ∧ (𝑠𝐾1) = (𝐻‘𝑠)))
41	40	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐾0) = (𝐺‘𝑠))
42	35, 41	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝐺‘𝑠))
43	42	mpteq2dva 5200	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐺‘𝑠)))
44		fovcdm 7559	. . . . . . . . . . 11 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
45	23, 44	mp3an3 1452	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
46	22, 45	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
47		eqidd 2730	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
48		cvmcn 35249	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
49	3, 48	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
50		eqid 2729	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
51	1, 50	cnf 23133	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
52	49, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
53	52	feqmptd 6929	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐹‘𝑥)))
54		fveq2 6858	. . . . . . . . 9 ⊢ (𝑥 = (𝑠𝐴0) → (𝐹‘𝑥) = (𝐹‘(𝑠𝐴0)))
55	46, 47, 53, 54	fmptco 7101	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))))
56	19, 50	cnf 23133	. . . . . . . . . 10 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶∪ 𝐽)
57	4, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:(0[,]1)⟶∪ 𝐽)
58	57	feqmptd 6929	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ (0[,]1) ↦ (𝐺‘𝑠)))
59	43, 55, 58	3eqtr4d 2774	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺)
60		cvmliftphtlem.0	. . . . . . 7 ⊢ (𝜑 → (0𝐴0) = 𝑃)
61	37	cnmptid 23548	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 𝑠) ∈ (II Cn II))
62	23	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0[,]1))
63	37, 37, 62	cnmptc 23549	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
64	37, 61, 63, 17	cnmpt12f 23553	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶))
65	1	cvmlift 35286	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
66	3, 4, 5, 6, 65	syl22anc 838	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
67		coeq2 5822	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
68	67	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺))
69		fveq1 6857	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0))
70		oveq1 7394	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠𝐴0) = (0𝐴0))
71		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))
72		ovex 7420	. . . . . . . . . . . . . 14 ⊢ (0𝐴0) ∈ V
73	70, 71, 72	fvmpt 6968	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0))
74	23, 73	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0)
75	69, 74	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = (0𝐴0))
76	75	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴0) = 𝑃))
77	68, 76	anbi12d 632	. . . . . . . . 9 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃)))
78	77	riota2 7369	. . . . . . . 8 ⊢ (((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶) ∧ ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃) ↔ (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
79	64, 66, 78	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃) ↔ (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
80	59, 60, 79	mpbi2and 712	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
81	2, 80	eqtrid 2776	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
82	19, 1	cnf 23133	. . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐶) → 𝑀:(0[,]1)⟶𝐵)
83	8, 82	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝐵)
84	83	feqmptd 6929	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)))
85	81, 84	eqtr3d 2766	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)))
86		mpteqb 6987	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠)))
87		ovexd 7422	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (𝑠𝐴0) ∈ V)
88	86, 87	mprg 3050	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠))
89	85, 88	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠))
90	89	r19.21bi 3229	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) = (𝑀‘𝑠))
91		1elunit 13431	. . . . . . . . . . . . . 14 ⊢ 1 ∈ (0[,]1)
92		opelxpi 5675	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
93	91, 92	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
94		fvco3 6960	. . . . . . . . . . . . 13 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
95	22, 93, 94	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
96	29	fveq1d 6860	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐾‘⟨𝑠, 1⟩))
97	95, 96	eqtr3d 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 1⟩)) = (𝐾‘⟨𝑠, 1⟩))
98		df-ov 7390	. . . . . . . . . . . 12 ⊢ (𝑠𝐴1) = (𝐴‘⟨𝑠, 1⟩)
99	98	fveq2i 6861	. . . . . . . . . . 11 ⊢ (𝐹‘(𝑠𝐴1)) = (𝐹‘(𝐴‘⟨𝑠, 1⟩))
100		df-ov 7390	. . . . . . . . . . 11 ⊢ (𝑠𝐾1) = (𝐾‘⟨𝑠, 1⟩)
101	97, 99, 100	3eqtr4g 2789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝑠𝐾1))
102	40	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐾1) = (𝐻‘𝑠))
103	101, 102	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝐻‘𝑠))
104	103	mpteq2dva 5200	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐻‘𝑠)))
105		fovcdm 7559	. . . . . . . . . . 11 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
106	91, 105	mp3an3 1452	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
107	22, 106	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
108		eqidd 2730	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
109		fveq2 6858	. . . . . . . . 9 ⊢ (𝑥 = (𝑠𝐴1) → (𝐹‘𝑥) = (𝐹‘(𝑠𝐴1)))
110	107, 108, 53, 109	fmptco 7101	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))))
111	19, 50	cnf 23133	. . . . . . . . . 10 ⊢ (𝐻 ∈ (II Cn 𝐽) → 𝐻:(0[,]1)⟶∪ 𝐽)
112	10, 111	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(0[,]1)⟶∪ 𝐽)
113	112	feqmptd 6929	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (0[,]1) ↦ (𝐻‘𝑠)))
114	104, 110, 113	3eqtr4d 2774	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻)
115		iiconn 24780	. . . . . . . . . . . . 13 ⊢ II ∈ Conn
116	115	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ Conn)
117		iinllyconn 35241	. . . . . . . . . . . . 13 ⊢ II ∈ 𝑛-Locally Conn
118	117	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ 𝑛-Locally Conn)
119	37, 63, 61, 17	cnmpt12f 23553	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) ∈ (II Cn 𝐶))
120		cvmtop1 35247	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
121	3, 120	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ Top)
122	1	toptopon 22804	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
123	121, 122	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
124		ffvelcdm 7053	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 0 ∈ (0[,]1)) → (𝑀‘0) ∈ 𝐵)
125	83, 23, 124	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘0) ∈ 𝐵)
126		cnconst2 23170	. . . . . . . . . . . . 13 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘0) ∈ 𝐵) → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
127	37, 123, 125, 126	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
128	4, 10, 11	phtpyi 24883	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐾𝑠) = (𝐺‘0) ∧ (1𝐾𝑠) = (𝐺‘1)))
129	128	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐾𝑠) = (𝐺‘0))
130		opelxpi 5675	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
131	23, 130	mpan 690	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (0[,]1) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
132		fvco3 6960	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
133	22, 131, 132	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
134	29	fveq1d 6860	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐾‘⟨0, 𝑠⟩))
135	133, 134	eqtr3d 2766	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨0, 𝑠⟩)) = (𝐾‘⟨0, 𝑠⟩))
136		df-ov 7390	. . . . . . . . . . . . . . . . . 18 ⊢ (0𝐴𝑠) = (𝐴‘⟨0, 𝑠⟩)
137	136	fveq2i 6861	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝐴‘⟨0, 𝑠⟩))
138		df-ov 7390	. . . . . . . . . . . . . . . . 17 ⊢ (0𝐾𝑠) = (𝐾‘⟨0, 𝑠⟩)
139	135, 137, 138	3eqtr4g 2789	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (0𝐾𝑠))
140	7	simp3d 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀‘0) = 𝑃)
141	140	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑀‘0) = 𝑃)
142	141	fveq2d 6862	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐹‘𝑃))
143	6	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑃) = (𝐺‘0))
144	142, 143	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐺‘0))
145	129, 139, 144	3eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝑀‘0)))
146	145	mpteq2dva 5200	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0))))
147		fconstmpt 5700	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × {(𝐹‘(𝑀‘0))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0)))
148	146, 147	eqtr4di 2782	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
149		fovcdm 7559	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
150	23, 149	mp3an2 1451	. . . . . . . . . . . . . . 15 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
151	22, 150	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
152		eqidd 2730	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)))
153		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (0𝐴𝑠) → (𝐹‘𝑥) = (𝐹‘(0𝐴𝑠)))
154	151, 152, 53, 153	fmptco 7101	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))))
155	52	ffnd 6689	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐵)
156		fcoconst 7106	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐵 ∧ (𝑀‘0) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
157	155, 125, 156	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
158	148, 154, 157	3eqtr4d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})))
159	60, 140	eqtr4d 2767	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0𝐴0) = (𝑀‘0))
160		oveq2 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0 → (0𝐴𝑠) = (0𝐴0))
161		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))
162	160, 161, 72	fvmpt 6968	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0))
163	23, 162	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0)
164		fvex 6871	. . . . . . . . . . . . . . 15 ⊢ (𝑀‘0) ∈ V
165	164	fvconst2 7178	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0))
166	23, 165	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0)
167	159, 163, 166	3eqtr4g 2789	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘0)})‘0))
168	1, 19, 3, 116, 118, 62, 119, 127, 158, 167	cvmliftmoi 35270	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = ((0[,]1) × {(𝑀‘0)}))
169		fconstmpt 5700	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝑀‘0)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0))
170	168, 169	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)))
171		mpteqb 6987	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0)))
172		ovexd 7422	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (0[,]1) → (0𝐴𝑠) ∈ V)
173	171, 172	mprg 3050	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
174	170, 173	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
175		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑠 = 1 → (0𝐴𝑠) = (0𝐴1))
176	175	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑠 = 1 → ((0𝐴𝑠) = (𝑀‘0) ↔ (0𝐴1) = (𝑀‘0)))
177	176	rspcv 3584	. . . . . . . . 9 ⊢ (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0) → (0𝐴1) = (𝑀‘0)))
178	91, 174, 177	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → (0𝐴1) = (𝑀‘0))
179	178, 140	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → (0𝐴1) = 𝑃)
180	91	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ (0[,]1))
181	37, 37, 180	cnmptc 23549	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 1) ∈ (II Cn II))
182	37, 61, 181, 17	cnmpt12f 23553	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶))
183	1	cvmlift 35286	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐻‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
184	3, 10, 5, 14, 183	syl22anc 838	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
185		coeq2 5822	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
186	185	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝐹 ∘ 𝑓) = 𝐻 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻))
187		fveq1 6857	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0))
188		oveq1 7394	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠𝐴1) = (0𝐴1))
189		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))
190		ovex 7420	. . . . . . . . . . . . . 14 ⊢ (0𝐴1) ∈ V
191	188, 189, 190	fvmpt 6968	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1))
192	23, 191	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1)
193	187, 192	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = (0𝐴1))
194	193	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴1) = 𝑃))
195	186, 194	anbi12d 632	. . . . . . . . 9 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃)))
196	195	riota2 7369	. . . . . . . 8 ⊢ (((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶) ∧ ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃) ↔ (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
197	182, 184, 196	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃) ↔ (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
198	114, 179, 197	mpbi2and 712	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
199	9, 198	eqtrid 2776	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
200	19, 1	cnf 23133	. . . . . . 7 ⊢ (𝑁 ∈ (II Cn 𝐶) → 𝑁:(0[,]1)⟶𝐵)
201	16, 200	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝐵)
202	201	feqmptd 6929	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)))
203	199, 202	eqtr3d 2766	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)))
204		mpteqb 6987	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(𝑠𝐴1) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠)))
205		ovexd 7422	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (𝑠𝐴1) ∈ V)
206	204, 205	mprg 3050	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠))
207	203, 206	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠))
208	207	r19.21bi 3229	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) = (𝑁‘𝑠))
209	174	r19.21bi 3229	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) = (𝑀‘0))
210	37, 181, 61, 17	cnmpt12f 23553	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) ∈ (II Cn 𝐶))
211		ffvelcdm 7053	. . . . . . . 8 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → (𝑀‘1) ∈ 𝐵)
212	83, 91, 211	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝑀‘1) ∈ 𝐵)
213		cnconst2 23170	. . . . . . 7 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘1) ∈ 𝐵) → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
214	37, 123, 212, 213	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
215		opelxpi 5675	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
216	91, 215	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
217		fvco3 6960	. . . . . . . . . . . . 13 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
218	22, 216, 217	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
219	29	fveq1d 6860	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐾‘⟨1, 𝑠⟩))
220	218, 219	eqtr3d 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨1, 𝑠⟩)) = (𝐾‘⟨1, 𝑠⟩))
221		df-ov 7390	. . . . . . . . . . . 12 ⊢ (1𝐴𝑠) = (𝐴‘⟨1, 𝑠⟩)
222	221	fveq2i 6861	. . . . . . . . . . 11 ⊢ (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝐴‘⟨1, 𝑠⟩))
223		df-ov 7390	. . . . . . . . . . 11 ⊢ (1𝐾𝑠) = (𝐾‘⟨1, 𝑠⟩)
224	220, 222, 223	3eqtr4g 2789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (1𝐾𝑠))
225	128	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐾𝑠) = (𝐺‘1))
226	7	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = 𝐺)
227	226	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹 ∘ 𝑀) = 𝐺)
228	227	fveq1d 6860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐺‘1))
229	83	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑀:(0[,]1)⟶𝐵)
230		fvco3 6960	. . . . . . . . . . . 12 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐹‘(𝑀‘1)))
231	229, 91, 230	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐹‘(𝑀‘1)))
232	228, 231	eqtr3d 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = (𝐹‘(𝑀‘1)))
233	224, 225, 232	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝑀‘1)))
234	233	mpteq2dva 5200	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1))))
235		fconstmpt 5700	. . . . . . . 8 ⊢ ((0[,]1) × {(𝐹‘(𝑀‘1))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1)))
236	234, 235	eqtr4di 2782	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
237		fovcdm 7559	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
238	91, 237	mp3an2 1451	. . . . . . . . 9 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
239	22, 238	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
240		eqidd 2730	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)))
241		fveq2 6858	. . . . . . . 8 ⊢ (𝑥 = (1𝐴𝑠) → (𝐹‘𝑥) = (𝐹‘(1𝐴𝑠)))
242	239, 240, 53, 241	fmptco 7101	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))))
243		fcoconst 7106	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐵 ∧ (𝑀‘1) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
244	155, 212, 243	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
245	236, 242, 244	3eqtr4d 2774	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})))
246		oveq1 7394	. . . . . . . . . 10 ⊢ (𝑠 = 1 → (𝑠𝐴0) = (1𝐴0))
247		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑠 = 1 → (𝑀‘𝑠) = (𝑀‘1))
248	246, 247	eqeq12d 2745	. . . . . . . . 9 ⊢ (𝑠 = 1 → ((𝑠𝐴0) = (𝑀‘𝑠) ↔ (1𝐴0) = (𝑀‘1)))
249	248	rspcv 3584	. . . . . . . 8 ⊢ (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠) → (1𝐴0) = (𝑀‘1)))
250	91, 89, 249	mpsyl 68	. . . . . . 7 ⊢ (𝜑 → (1𝐴0) = (𝑀‘1))
251		oveq2 7395	. . . . . . . . 9 ⊢ (𝑠 = 0 → (1𝐴𝑠) = (1𝐴0))
252		eqid 2729	. . . . . . . . 9 ⊢ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))
253		ovex 7420	. . . . . . . . 9 ⊢ (1𝐴0) ∈ V
254	251, 252, 253	fvmpt 6968	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0))
255	23, 254	ax-mp 5	. . . . . . 7 ⊢ ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0)
256		fvex 6871	. . . . . . . . 9 ⊢ (𝑀‘1) ∈ V
257	256	fvconst2 7178	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1))
258	23, 257	ax-mp 5	. . . . . . 7 ⊢ (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1)
259	250, 255, 258	3eqtr4g 2789	. . . . . 6 ⊢ (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘1)})‘0))
260	1, 19, 3, 116, 118, 62, 210, 214, 245, 259	cvmliftmoi 35270	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = ((0[,]1) × {(𝑀‘1)}))
261		fconstmpt 5700	. . . . 5 ⊢ ((0[,]1) × {(𝑀‘1)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1))
262	260, 261	eqtrdi 2780	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)))
263		mpteqb 6987	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(1𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1)))
264		ovexd 7422	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (1𝐴𝑠) ∈ V)
265	263, 264	mprg 3050	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
266	262, 265	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
267	266	r19.21bi 3229	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) = (𝑀‘1))
268	8, 16, 17, 90, 208, 209, 267	isphtpy2d 24886	1 ⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃!wreu 3352 Vcvv 3447 {csn 4589 ⟨cop 4595 ∪ cuni 4871 ↦ cmpt 5188 × cxp 5636 ∘ ccom 5642 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 ℩crio 7343 (class class class)co 7387 0cc0 11068 1c1 11069 [,]cicc 13309 Topctop 22780 TopOnctopon 22797 Cn ccn 23111 Conncconn 23298 𝑛-Locally cnlly 23352 ×_t ctx 23447 IIcii 24768 Htpy chtpy 24866 PHtpycphtpy 24867 CovMap ccvm 35242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-ec 8673 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-cn 23114 df-cnp 23115 df-cmp 23274 df-conn 23299 df-lly 23353 df-nlly 23354 df-tx 23449 df-hmeo 23642 df-xms 24208 df-ms 24209 df-tms 24210 df-ii 24770 df-cncf 24771 df-htpy 24869 df-phtpy 24870 df-phtpc 24891 df-pconn 35208 df-sconn 35209 df-cvm 35243

This theorem is referenced by: cvmliftpht 35305

W3C validator