cvmliftphtlem - Mathbox for Mario Carneiro

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Theorem cvmliftphtlem 35605

Description: Lemma for cvmliftpht 35606. (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 35589	. . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
8	7	simp1d 1151	. 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 25016	. . . . . 6 ⊢ (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
13	12	simpld 497	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0))
14	6, 13	eqtrd 2787	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
15	1, 9, 3, 10, 5, 14	cvmliftiota 35589	. . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
16	15	simp1d 1151	. 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶))
17		cvmliftphtlem.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ((II ×_t II) Cn 𝐶))
18		iitop 24911	. . . . . . . . . . . . . . . 16 ⊢ II ∈ Top
19		iiuni 24912	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) = ∪ II
20	18, 18, 19, 19	txunii 23622	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
21	20, 1	cnf 23275	. . . . . . . . . . . . . 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 13459	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0[,]1)
24		opelxpi 5673	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
25	23, 24	mpan2 699	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
26		fvco3 6952	. . . . . . . . . . . . 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 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
28		cvmliftphtlem.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∘ 𝐴) = 𝐾)
29	28	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹 ∘ 𝐴) = 𝐾)
30	29	fveq1d 6854	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 0⟩) = (𝐾‘⟨𝑠, 0⟩))
31	27, 30	eqtr3d 2789	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 0⟩)) = (𝐾‘⟨𝑠, 0⟩))
32		df-ov 7384	. . . . . . . . . . . 12 ⊢ (𝑠𝐴0) = (𝐴‘⟨𝑠, 0⟩)
33	32	fveq2i 6855	. . . . . . . . . . 11 ⊢ (𝐹‘(𝑠𝐴0)) = (𝐹‘(𝐴‘⟨𝑠, 0⟩))
34		df-ov 7384	. . . . . . . . . . 11 ⊢ (𝑠𝐾0) = (𝐾‘⟨𝑠, 0⟩)
35	31, 33, 34	3eqtr4g 2812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝑠𝐾0))
36		iitopon 24910	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
38	4, 10	phtpyhtpy 25013	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ (𝐺(II Htpy 𝐽)𝐻))
39	38, 11	sseldd 3928	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐺(II Htpy 𝐽)𝐻))
40	37, 4, 10, 39	htpyi 25005	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐾0) = (𝐺‘𝑠) ∧ (𝑠𝐾1) = (𝐻‘𝑠)))
41	40	simpld 497	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐾0) = (𝐺‘𝑠))
42	35, 41	eqtrd 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝐺‘𝑠))
43	42	mpteq2dva 5183	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐺‘𝑠)))
44		fovcdm 7551	. . . . . . . . . . 11 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
45	23, 44	mp3an3 1461	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
46	22, 45	sylan 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
47		eqidd 2753	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
48		cvmcn 35550	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
49	3, 48	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
50		eqid 2752	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
51	1, 50	cnf 23275	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
52	49, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
53	52	feqmptd 6920	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐹‘𝑥)))
54		fveq2 6852	. . . . . . . . 9 ⊢ (𝑥 = (𝑠𝐴0) → (𝐹‘𝑥) = (𝐹‘(𝑠𝐴0)))
55	46, 47, 53, 54	fmptco 7096	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))))
56	19, 50	cnf 23275	. . . . . . . . . 10 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶∪ 𝐽)
57	4, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:(0[,]1)⟶∪ 𝐽)
58	57	feqmptd 6920	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ (0[,]1) ↦ (𝐺‘𝑠)))
59	43, 55, 58	3eqtr4d 2797	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺)
60		cvmliftphtlem.0	. . . . . . 7 ⊢ (𝜑 → (0𝐴0) = 𝑃)
61	37	cnmptid 23690	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 𝑠) ∈ (II Cn II))
62	23	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0[,]1))
63	37, 37, 62	cnmptc 23691	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
64	37, 61, 63, 17	cnmpt12f 23695	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶))
65	1	cvmlift 35587	. . . . . . . . 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 847	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
67		coeq2 5819	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
68	67	eqeq1d 2754	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺))
69		fveq1 6851	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0))
70		oveq1 7388	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠𝐴0) = (0𝐴0))
71		eqid 2752	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))
72		ovex 7414	. . . . . . . . . . . . . 14 ⊢ (0𝐴0) ∈ V
73	70, 71, 72	fvmpt 6960	. . . . . . . . . . . . 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 2803	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = (0𝐴0))
76	75	eqeq1d 2754	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴0) = 𝑃))
77	68, 76	anbi12d 640	. . . . . . . . 9 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃)))
78	77	riota2 7363	. . . . . . . 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 592	. . . . . . 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 720	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
81	2, 80	eqtrid 2799	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
82	19, 1	cnf 23275	. . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐶) → 𝑀:(0[,]1)⟶𝐵)
83	8, 82	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝐵)
84	83	feqmptd 6920	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)))
85	81, 84	eqtr3d 2789	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)))
86		mpteqb 6980	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠)))
87		ovexd 7416	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (𝑠𝐴0) ∈ V)
88	86, 87	mprg 3072	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠))
89	85, 88	sylib 220	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠))
90	89	r19.21bi 3244	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) = (𝑀‘𝑠))
91		1elunit 13460	. . . . . . . . . . . . . 14 ⊢ 1 ∈ (0[,]1)
92		opelxpi 5673	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
93	91, 92	mpan2 699	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
94		fvco3 6952	. . . . . . . . . . . . 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 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
96	29	fveq1d 6854	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐾‘⟨𝑠, 1⟩))
97	95, 96	eqtr3d 2789	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 1⟩)) = (𝐾‘⟨𝑠, 1⟩))
98		df-ov 7384	. . . . . . . . . . . 12 ⊢ (𝑠𝐴1) = (𝐴‘⟨𝑠, 1⟩)
99	98	fveq2i 6855	. . . . . . . . . . 11 ⊢ (𝐹‘(𝑠𝐴1)) = (𝐹‘(𝐴‘⟨𝑠, 1⟩))
100		df-ov 7384	. . . . . . . . . . 11 ⊢ (𝑠𝐾1) = (𝐾‘⟨𝑠, 1⟩)
101	97, 99, 100	3eqtr4g 2812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝑠𝐾1))
102	40	simprd 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐾1) = (𝐻‘𝑠))
103	101, 102	eqtrd 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝐻‘𝑠))
104	103	mpteq2dva 5183	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐻‘𝑠)))
105		fovcdm 7551	. . . . . . . . . . 11 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
106	91, 105	mp3an3 1461	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
107	22, 106	sylan 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
108		eqidd 2753	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
109		fveq2 6852	. . . . . . . . 9 ⊢ (𝑥 = (𝑠𝐴1) → (𝐹‘𝑥) = (𝐹‘(𝑠𝐴1)))
110	107, 108, 53, 109	fmptco 7096	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))))
111	19, 50	cnf 23275	. . . . . . . . . 10 ⊢ (𝐻 ∈ (II Cn 𝐽) → 𝐻:(0[,]1)⟶∪ 𝐽)
112	10, 111	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(0[,]1)⟶∪ 𝐽)
113	112	feqmptd 6920	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (0[,]1) ↦ (𝐻‘𝑠)))
114	104, 110, 113	3eqtr4d 2797	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻)
115		iiconn 24918	. . . . . . . . . . . . 13 ⊢ II ∈ Conn
116	115	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ Conn)
117		iinllyconn 35542	. . . . . . . . . . . . 13 ⊢ II ∈ 𝑛-Locally Conn
118	117	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ 𝑛-Locally Conn)
119	37, 63, 61, 17	cnmpt12f 23695	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) ∈ (II Cn 𝐶))
120		cvmtop1 35548	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
121	3, 120	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ Top)
122	1	toptopon 22946	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
123	121, 122	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
124		ffvelcdm 7047	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 0 ∈ (0[,]1)) → (𝑀‘0) ∈ 𝐵)
125	83, 23, 124	sylancl 594	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘0) ∈ 𝐵)
126		cnconst2 23312	. . . . . . . . . . . . 13 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘0) ∈ 𝐵) → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
127	37, 123, 125, 126	syl3anc 1382	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
128	4, 10, 11	phtpyi 25015	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐾𝑠) = (𝐺‘0) ∧ (1𝐾𝑠) = (𝐺‘1)))
129	128	simpld 497	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐾𝑠) = (𝐺‘0))
130		opelxpi 5673	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
131	23, 130	mpan 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (0[,]1) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
132		fvco3 6952	. . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
134	29	fveq1d 6854	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐾‘⟨0, 𝑠⟩))
135	133, 134	eqtr3d 2789	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨0, 𝑠⟩)) = (𝐾‘⟨0, 𝑠⟩))
136		df-ov 7384	. . . . . . . . . . . . . . . . . 18 ⊢ (0𝐴𝑠) = (𝐴‘⟨0, 𝑠⟩)
137	136	fveq2i 6855	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝐴‘⟨0, 𝑠⟩))
138		df-ov 7384	. . . . . . . . . . . . . . . . 17 ⊢ (0𝐾𝑠) = (𝐾‘⟨0, 𝑠⟩)
139	135, 137, 138	3eqtr4g 2812	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (0𝐾𝑠))
140	7	simp3d 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀‘0) = 𝑃)
141	140	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑀‘0) = 𝑃)
142	141	fveq2d 6856	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐹‘𝑃))
143	6	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑃) = (𝐺‘0))
144	142, 143	eqtrd 2787	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐺‘0))
145	129, 139, 144	3eqtr4d 2797	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝑀‘0)))
146	145	mpteq2dva 5183	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0))))
147		fconstmpt 5698	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × {(𝐹‘(𝑀‘0))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0)))
148	146, 147	eqtr4di 2805	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
149		fovcdm 7551	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
150	23, 149	mp3an2 1460	. . . . . . . . . . . . . . 15 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
151	22, 150	sylan 588	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
152		eqidd 2753	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)))
153		fveq2 6852	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (0𝐴𝑠) → (𝐹‘𝑥) = (𝐹‘(0𝐴𝑠)))
154	151, 152, 53, 153	fmptco 7096	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))))
155	52	ffnd 6677	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐵)
156		fcoconst 7101	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐵 ∧ (𝑀‘0) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
157	155, 125, 156	syl2anc 592	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
158	148, 154, 157	3eqtr4d 2797	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})))
159	60, 140	eqtr4d 2790	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0𝐴0) = (𝑀‘0))
160		oveq2 7389	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0 → (0𝐴𝑠) = (0𝐴0))
161		eqid 2752	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))
162	160, 161, 72	fvmpt 6960	. . . . . . . . . . . . . 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 6865	. . . . . . . . . . . . . . 15 ⊢ (𝑀‘0) ∈ V
165	164	fvconst2 7173	. . . . . . . . . . . . . 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 2812	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘0)})‘0))
168	1, 19, 3, 116, 118, 62, 119, 127, 158, 167	cvmliftmoi 35571	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = ((0[,]1) × {(𝑀‘0)}))
169		fconstmpt 5698	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝑀‘0)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0))
170	168, 169	eqtrdi 2803	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)))
171		mpteqb 6980	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0)))
172		ovexd 7416	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (0[,]1) → (0𝐴𝑠) ∈ V)
173	171, 172	mprg 3072	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
174	170, 173	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
175		oveq2 7389	. . . . . . . . . . 11 ⊢ (𝑠 = 1 → (0𝐴𝑠) = (0𝐴1))
176	175	eqeq1d 2754	. . . . . . . . . 10 ⊢ (𝑠 = 1 → ((0𝐴𝑠) = (𝑀‘0) ↔ (0𝐴1) = (𝑀‘0)))
177	176	rspcv 3568	. . . . . . . . 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 2787	. . . . . . 7 ⊢ (𝜑 → (0𝐴1) = 𝑃)
180	91	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ (0[,]1))
181	37, 37, 180	cnmptc 23691	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 1) ∈ (II Cn II))
182	37, 61, 181, 17	cnmpt12f 23695	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶))
183	1	cvmlift 35587	. . . . . . . . 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 847	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
185		coeq2 5819	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
186	185	eqeq1d 2754	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝐹 ∘ 𝑓) = 𝐻 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻))
187		fveq1 6851	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0))
188		oveq1 7388	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠𝐴1) = (0𝐴1))
189		eqid 2752	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))
190		ovex 7414	. . . . . . . . . . . . . 14 ⊢ (0𝐴1) ∈ V
191	188, 189, 190	fvmpt 6960	. . . . . . . . . . . . 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 2803	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = (0𝐴1))
194	193	eqeq1d 2754	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴1) = 𝑃))
195	186, 194	anbi12d 640	. . . . . . . . 9 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃)))
196	195	riota2 7363	. . . . . . . 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 592	. . . . . . 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 720	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
199	9, 198	eqtrid 2799	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
200	19, 1	cnf 23275	. . . . . . 7 ⊢ (𝑁 ∈ (II Cn 𝐶) → 𝑁:(0[,]1)⟶𝐵)
201	16, 200	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝐵)
202	201	feqmptd 6920	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)))
203	199, 202	eqtr3d 2789	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)))
204		mpteqb 6980	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(𝑠𝐴1) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠)))
205		ovexd 7416	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (𝑠𝐴1) ∈ V)
206	204, 205	mprg 3072	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠))
207	203, 206	sylib 220	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠))
208	207	r19.21bi 3244	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) = (𝑁‘𝑠))
209	174	r19.21bi 3244	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) = (𝑀‘0))
210	37, 181, 61, 17	cnmpt12f 23695	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) ∈ (II Cn 𝐶))
211		ffvelcdm 7047	. . . . . . . 8 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → (𝑀‘1) ∈ 𝐵)
212	83, 91, 211	sylancl 594	. . . . . . 7 ⊢ (𝜑 → (𝑀‘1) ∈ 𝐵)
213		cnconst2 23312	. . . . . . 7 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘1) ∈ 𝐵) → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
214	37, 123, 212, 213	syl3anc 1382	. . . . . 6 ⊢ (𝜑 → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
215		opelxpi 5673	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
216	91, 215	mpan 698	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
217		fvco3 6952	. . . . . . . . . . . . 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 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
219	29	fveq1d 6854	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐾‘⟨1, 𝑠⟩))
220	218, 219	eqtr3d 2789	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨1, 𝑠⟩)) = (𝐾‘⟨1, 𝑠⟩))
221		df-ov 7384	. . . . . . . . . . . 12 ⊢ (1𝐴𝑠) = (𝐴‘⟨1, 𝑠⟩)
222	221	fveq2i 6855	. . . . . . . . . . 11 ⊢ (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝐴‘⟨1, 𝑠⟩))
223		df-ov 7384	. . . . . . . . . . 11 ⊢ (1𝐾𝑠) = (𝐾‘⟨1, 𝑠⟩)
224	220, 222, 223	3eqtr4g 2812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (1𝐾𝑠))
225	128	simprd 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐾𝑠) = (𝐺‘1))
226	7	simp2d 1152	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = 𝐺)
227	226	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹 ∘ 𝑀) = 𝐺)
228	227	fveq1d 6854	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐺‘1))
229	83	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑀:(0[,]1)⟶𝐵)
230		fvco3 6952	. . . . . . . . . . . 12 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐹‘(𝑀‘1)))
231	229, 91, 230	sylancl 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐹‘(𝑀‘1)))
232	228, 231	eqtr3d 2789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = (𝐹‘(𝑀‘1)))
233	224, 225, 232	3eqtrd 2791	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝑀‘1)))
234	233	mpteq2dva 5183	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1))))
235		fconstmpt 5698	. . . . . . . 8 ⊢ ((0[,]1) × {(𝐹‘(𝑀‘1))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1)))
236	234, 235	eqtr4di 2805	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
237		fovcdm 7551	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
238	91, 237	mp3an2 1460	. . . . . . . . 9 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
239	22, 238	sylan 588	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
240		eqidd 2753	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)))
241		fveq2 6852	. . . . . . . 8 ⊢ (𝑥 = (1𝐴𝑠) → (𝐹‘𝑥) = (𝐹‘(1𝐴𝑠)))
242	239, 240, 53, 241	fmptco 7096	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))))
243		fcoconst 7101	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐵 ∧ (𝑀‘1) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
244	155, 212, 243	syl2anc 592	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
245	236, 242, 244	3eqtr4d 2797	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})))
246		oveq1 7388	. . . . . . . . . 10 ⊢ (𝑠 = 1 → (𝑠𝐴0) = (1𝐴0))
247		fveq2 6852	. . . . . . . . . 10 ⊢ (𝑠 = 1 → (𝑀‘𝑠) = (𝑀‘1))
248	246, 247	eqeq12d 2768	. . . . . . . . 9 ⊢ (𝑠 = 1 → ((𝑠𝐴0) = (𝑀‘𝑠) ↔ (1𝐴0) = (𝑀‘1)))
249	248	rspcv 3568	. . . . . . . 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 7389	. . . . . . . . 9 ⊢ (𝑠 = 0 → (1𝐴𝑠) = (1𝐴0))
252		eqid 2752	. . . . . . . . 9 ⊢ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))
253		ovex 7414	. . . . . . . . 9 ⊢ (1𝐴0) ∈ V
254	251, 252, 253	fvmpt 6960	. . . . . . . 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 6865	. . . . . . . . 9 ⊢ (𝑀‘1) ∈ V
257	256	fvconst2 7173	. . . . . . . 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 2812	. . . . . 6 ⊢ (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘1)})‘0))
260	1, 19, 3, 116, 118, 62, 210, 214, 245, 259	cvmliftmoi 35571	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = ((0[,]1) × {(𝑀‘1)}))
261		fconstmpt 5698	. . . . 5 ⊢ ((0[,]1) × {(𝑀‘1)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1))
262	260, 261	eqtrdi 2803	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)))
263		mpteqb 6980	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(1𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1)))
264		ovexd 7416	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (1𝐴𝑠) ∈ V)
265	263, 264	mprg 3072	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
266	262, 265	sylib 220	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
267	266	r19.21bi 3244	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) = (𝑀‘1))
268	8, 16, 17, 90, 208, 209, 267	isphtpy2d 25018	1 ⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ∀wral 3066 ∃!wreu 3355 Vcvv 3444 {csn 4572 ⟨cop 4578 ∪ cuni 4855 ↦ cmpt 5171 × cxp 5634 ∘ ccom 5640 Fn wfn 6501 ⟶wf 6502 ‘cfv 6506 ℩crio 7337 (class class class)co 7381 0cc0 11059 1c1 11060 [,]cicc 13338 Topctop 22922 TopOnctopon 22939 Cn ccn 23253 Conncconn 23440 𝑛-Locally cnlly 23494 ×_t ctx 23589 IIcii 24906 Htpy chtpy 24998 PHtpycphtpy 24999 CovMap ccvm 35543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-ec 8664 df-map 8794 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-seq 14001 df-exp 14061 df-hash 14330 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-clim 15487 df-sum 15686 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 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 24349 df-ms 24350 df-tms 24351 df-ii 24908 df-cncf 24909 df-htpy 25001 df-phtpy 25002 df-phtpc 25023 df-pconn 35509 df-sconn 35510 df-cvm 35544

This theorem is referenced by: cvmliftpht 35606

W3C validator