cvmliftphtlem - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftphtlem		Structured version Visualization version GIF version

Theorem cvmliftphtlem 30553

Description: Lemma for cvmliftpht 30554. (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 30537	. . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
8	7	simp1d 1066	. 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 22592	. . . . . 6 ⊢ (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
13	12	simpld 474	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0))
14	6, 13	eqtrd 2644	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
15	1, 9, 3, 10, 5, 14	cvmliftiota 30537	. . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
16	15	simp1d 1066	. 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶))
17		cvmliftphtlem.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ((II ×_t II) Cn 𝐶))
18		iitop 22491	. . . . . . . . . . . . . . . 16 ⊢ II ∈ Top
19		iiuni 22492	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) = ∪ II
20	18, 18, 19, 19	txunii 21206	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
21	20, 1	cnf 20860	. . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0[,]1)
24		opelxpi 5072	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
25	23, 24	mpan2 703	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
26		fvco3 6185	. . . . . . . . . . . . 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 493	. . . . . . . . . . . 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 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 0⟩) = (𝐾‘⟨𝑠, 0⟩))
31	27, 30	eqtr3d 2646	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 0⟩)) = (𝐾‘⟨𝑠, 0⟩))
32		df-ov 6552	. . . . . . . . . . . 12 ⊢ (𝑠𝐴0) = (𝐴‘⟨𝑠, 0⟩)
33	32	fveq2i 6106	. . . . . . . . . . 11 ⊢ (𝐹‘(𝑠𝐴0)) = (𝐹‘(𝐴‘⟨𝑠, 0⟩))
34		df-ov 6552	. . . . . . . . . . 11 ⊢ (𝑠𝐾0) = (𝐾‘⟨𝑠, 0⟩)
35	31, 33, 34	3eqtr4g 2669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝑠𝐾0))
36		iitopon 22490	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
38	4, 10	phtpyhtpy 22589	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ (𝐺(II Htpy 𝐽)𝐻))
39	38, 11	sseldd 3569	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐺(II Htpy 𝐽)𝐻))
40	37, 4, 10, 39	htpyi 22581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐾0) = (𝐺‘𝑠) ∧ (𝑠𝐾1) = (𝐻‘𝑠)))
41	40	simpld 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐾0) = (𝐺‘𝑠))
42	35, 41	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝐺‘𝑠))
43	42	mpteq2dva 4672	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐺‘𝑠)))
44		fovrn 6702	. . . . . . . . . . 11 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
45	23, 44	mp3an3 1405	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
46	22, 45	sylan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
47		eqidd 2611	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
48		cvmcn 30498	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
49	3, 48	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
50		eqid 2610	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
51	1, 50	cnf 20860	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
52	49, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
53	52	feqmptd 6159	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐹‘𝑥)))
54		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = (𝑠𝐴0) → (𝐹‘𝑥) = (𝐹‘(𝑠𝐴0)))
55	46, 47, 53, 54	fmptco 6303	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))))
56	19, 50	cnf 20860	. . . . . . . . . 10 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶∪ 𝐽)
57	4, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:(0[,]1)⟶∪ 𝐽)
58	57	feqmptd 6159	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ (0[,]1) ↦ (𝐺‘𝑠)))
59	43, 55, 58	3eqtr4d 2654	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺)
60		cvmliftphtlem.0	. . . . . . 7 ⊢ (𝜑 → (0𝐴0) = 𝑃)
61	37	cnmptid 21274	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 𝑠) ∈ (II Cn II))
62	23	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0[,]1))
63	37, 37, 62	cnmptc 21275	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
64	37, 61, 63, 17	cnmpt12f 21279	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶))
65	1	cvmlift 30535	. . . . . . . . 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 1319	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
67		coeq2 5202	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
68	67	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺))
69		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0))
70		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠𝐴0) = (0𝐴0))
71		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))
72		ovex 6577	. . . . . . . . . . . . . 14 ⊢ (0𝐴0) ∈ V
73	70, 71, 72	fvmpt 6191	. . . . . . . . . . . . 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	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = (0𝐴0))
76	75	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴0) = 𝑃))
77	68, 76	anbi12d 743	. . . . . . . . 9 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃)))
78	77	riota2 6533	. . . . . . . 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 691	. . . . . . 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 958	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
81	2, 80	syl5eq 2656	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
82	19, 1	cnf 20860	. . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐶) → 𝑀:(0[,]1)⟶𝐵)
83	8, 82	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝐵)
84	83	feqmptd 6159	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)))
85	81, 84	eqtr3d 2646	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)))
86		mpteqb 6207	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠)))
87		ovex 6577	. . . . . 6 ⊢ (𝑠𝐴0) ∈ V
88	87	a1i 11	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (𝑠𝐴0) ∈ V)
89	86, 88	mprg 2910	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠))
90	85, 89	sylib 207	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠))
91	90	r19.21bi 2916	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴0) = (𝑀‘𝑠))
92		1elunit 12162	. . . . . . . . . . . . . 14 ⊢ 1 ∈ (0[,]1)
93		opelxpi 5072	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
94	92, 93	mpan2 703	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
95		fvco3 6185	. . . . . . . . . . . . 13 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
96	22, 94, 95	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
97	29	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨𝑠, 1⟩) = (𝐾‘⟨𝑠, 1⟩))
98	96, 97	eqtr3d 2646	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 1⟩)) = (𝐾‘⟨𝑠, 1⟩))
99		df-ov 6552	. . . . . . . . . . . 12 ⊢ (𝑠𝐴1) = (𝐴‘⟨𝑠, 1⟩)
100	99	fveq2i 6106	. . . . . . . . . . 11 ⊢ (𝐹‘(𝑠𝐴1)) = (𝐹‘(𝐴‘⟨𝑠, 1⟩))
101		df-ov 6552	. . . . . . . . . . 11 ⊢ (𝑠𝐾1) = (𝐾‘⟨𝑠, 1⟩)
102	98, 100, 101	3eqtr4g 2669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝑠𝐾1))
103	40	simprd 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐾1) = (𝐻‘𝑠))
104	102, 103	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝐻‘𝑠))
105	104	mpteq2dva 4672	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐻‘𝑠)))
106		fovrn 6702	. . . . . . . . . . 11 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
107	92, 106	mp3an3 1405	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
108	22, 107	sylan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
109		eqidd 2611	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
110		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = (𝑠𝐴1) → (𝐹‘𝑥) = (𝐹‘(𝑠𝐴1)))
111	108, 109, 53, 110	fmptco 6303	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))))
112	19, 50	cnf 20860	. . . . . . . . . 10 ⊢ (𝐻 ∈ (II Cn 𝐽) → 𝐻:(0[,]1)⟶∪ 𝐽)
113	10, 112	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(0[,]1)⟶∪ 𝐽)
114	113	feqmptd 6159	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (0[,]1) ↦ (𝐻‘𝑠)))
115	105, 111, 114	3eqtr4d 2654	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻)
116		iicon 22498	. . . . . . . . . . . . 13 ⊢ II ∈ Con
117	116	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ Con)
118		iinllycon 30490	. . . . . . . . . . . . 13 ⊢ II ∈ 𝑛-Locally Con
119	118	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ 𝑛-Locally Con)
120	37, 63, 61, 17	cnmpt12f 21279	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) ∈ (II Cn 𝐶))
121		cvmtop1 30496	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
122	3, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ Top)
123	1	toptopon 20548	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
124	122, 123	sylib 207	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
125		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 0 ∈ (0[,]1)) → (𝑀‘0) ∈ 𝐵)
126	83, 23, 125	sylancl 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘0) ∈ 𝐵)
127		cnconst2 20897	. . . . . . . . . . . . 13 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘0) ∈ 𝐵) → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
128	37, 124, 126, 127	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
129	4, 10, 11	phtpyi 22591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐾𝑠) = (𝐺‘0) ∧ (1𝐾𝑠) = (𝐺‘1)))
130	129	simpld 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐾𝑠) = (𝐺‘0))
131		opelxpi 5072	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
132	23, 131	mpan 702	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (0[,]1) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
133		fvco3 6185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
134	22, 132, 133	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
135	29	fveq1d 6105	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨0, 𝑠⟩) = (𝐾‘⟨0, 𝑠⟩))
136	134, 135	eqtr3d 2646	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨0, 𝑠⟩)) = (𝐾‘⟨0, 𝑠⟩))
137		df-ov 6552	. . . . . . . . . . . . . . . . . 18 ⊢ (0𝐴𝑠) = (𝐴‘⟨0, 𝑠⟩)
138	137	fveq2i 6106	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝐴‘⟨0, 𝑠⟩))
139		df-ov 6552	. . . . . . . . . . . . . . . . 17 ⊢ (0𝐾𝑠) = (𝐾‘⟨0, 𝑠⟩)
140	136, 138, 139	3eqtr4g 2669	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (0𝐾𝑠))
141	7	simp3d 1068	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀‘0) = 𝑃)
142	141	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑀‘0) = 𝑃)
143	142	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐹‘𝑃))
144	6	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑃) = (𝐺‘0))
145	143, 144	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐺‘0))
146	130, 140, 145	3eqtr4d 2654	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝑀‘0)))
147	146	mpteq2dva 4672	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0))))
148		fconstmpt 5085	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × {(𝐹‘(𝑀‘0))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0)))
149	147, 148	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
150		fovrn 6702	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
151	23, 150	mp3an2 1404	. . . . . . . . . . . . . . 15 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
152	22, 151	sylan 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
153		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)))
154		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (0𝐴𝑠) → (𝐹‘𝑥) = (𝐹‘(0𝐴𝑠)))
155	152, 153, 53, 154	fmptco 6303	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))))
156		ffn 5958	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵⟶∪ 𝐽 → 𝐹 Fn 𝐵)
157	52, 156	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐵)
158		fcoconst 6307	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐵 ∧ (𝑀‘0) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
159	157, 126, 158	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
160	149, 155, 159	3eqtr4d 2654	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})))
161	60, 141	eqtr4d 2647	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0𝐴0) = (𝑀‘0))
162		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0 → (0𝐴𝑠) = (0𝐴0))
163		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))
164	162, 163, 72	fvmpt 6191	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0))
165	23, 164	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0)
166		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝑀‘0) ∈ V
167	166	fvconst2 6374	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0))
168	23, 167	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0)
169	161, 165, 168	3eqtr4g 2669	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘0)})‘0))
170	1, 19, 3, 117, 119, 62, 120, 128, 160, 169	cvmliftmoi 30519	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = ((0[,]1) × {(𝑀‘0)}))
171		fconstmpt 5085	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝑀‘0)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0))
172	170, 171	syl6eq 2660	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)))
173		mpteqb 6207	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0)))
174		ovex 6577	. . . . . . . . . . . 12 ⊢ (0𝐴𝑠) ∈ V
175	174	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (0[,]1) → (0𝐴𝑠) ∈ V)
176	173, 175	mprg 2910	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
177	172, 176	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
178		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑠 = 1 → (0𝐴𝑠) = (0𝐴1))
179	178	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑠 = 1 → ((0𝐴𝑠) = (𝑀‘0) ↔ (0𝐴1) = (𝑀‘0)))
180	179	rspcv 3278	. . . . . . . . 9 ⊢ (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0) → (0𝐴1) = (𝑀‘0)))
181	92, 177, 180	mpsyl 66	. . . . . . . 8 ⊢ (𝜑 → (0𝐴1) = (𝑀‘0))
182	181, 141	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (0𝐴1) = 𝑃)
183	92	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ (0[,]1))
184	37, 37, 183	cnmptc 21275	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ 1) ∈ (II Cn II))
185	37, 61, 184, 17	cnmpt12f 21279	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶))
186	1	cvmlift 30535	. . . . . . . . 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))
187	3, 10, 5, 14, 186	syl22anc 1319	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
188		coeq2 5202	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
189	188	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝐹 ∘ 𝑓) = 𝐻 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻))
190		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0))
191		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0 → (𝑠𝐴1) = (0𝐴1))
192		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))
193		ovex 6577	. . . . . . . . . . . . . 14 ⊢ (0𝐴1) ∈ V
194	191, 192, 193	fvmpt 6191	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1))
195	23, 194	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1)
196	190, 195	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = (0𝐴1))
197	196	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴1) = 𝑃))
198	189, 197	anbi12d 743	. . . . . . . . 9 ⊢ (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃)))
199	198	riota2 6533	. . . . . . . 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))))
200	185, 187, 199	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃) ↔ (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
201	115, 182, 200	mpbi2and 958	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
202	9, 201	syl5eq 2656	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
203	19, 1	cnf 20860	. . . . . . 7 ⊢ (𝑁 ∈ (II Cn 𝐶) → 𝑁:(0[,]1)⟶𝐵)
204	16, 203	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝐵)
205	204	feqmptd 6159	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)))
206	202, 205	eqtr3d 2646	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)))
207		mpteqb 6207	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(𝑠𝐴1) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠)))
208		ovex 6577	. . . . . 6 ⊢ (𝑠𝐴1) ∈ V
209	208	a1i 11	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (𝑠𝐴1) ∈ V)
210	207, 209	mprg 2910	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁‘𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠))
211	206, 210	sylib 207	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁‘𝑠))
212	211	r19.21bi 2916	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐴1) = (𝑁‘𝑠))
213	177	r19.21bi 2916	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) = (𝑀‘0))
214	37, 184, 61, 17	cnmpt12f 21279	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) ∈ (II Cn 𝐶))
215		ffvelrn 6265	. . . . . . . 8 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → (𝑀‘1) ∈ 𝐵)
216	83, 92, 215	sylancl 693	. . . . . . 7 ⊢ (𝜑 → (𝑀‘1) ∈ 𝐵)
217		cnconst2 20897	. . . . . . 7 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘1) ∈ 𝐵) → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
218	37, 124, 216, 217	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
219		opelxpi 5072	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
220	92, 219	mpan 702	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (0[,]1) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
221		fvco3 6185	. . . . . . . . . . . . 13 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
222	22, 220, 221	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
223	29	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐴)‘⟨1, 𝑠⟩) = (𝐾‘⟨1, 𝑠⟩))
224	222, 223	eqtr3d 2646	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨1, 𝑠⟩)) = (𝐾‘⟨1, 𝑠⟩))
225		df-ov 6552	. . . . . . . . . . . 12 ⊢ (1𝐴𝑠) = (𝐴‘⟨1, 𝑠⟩)
226	225	fveq2i 6106	. . . . . . . . . . 11 ⊢ (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝐴‘⟨1, 𝑠⟩))
227		df-ov 6552	. . . . . . . . . . 11 ⊢ (1𝐾𝑠) = (𝐾‘⟨1, 𝑠⟩)
228	224, 226, 227	3eqtr4g 2669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (1𝐾𝑠))
229	129	simprd 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐾𝑠) = (𝐺‘1))
230	7	simp2d 1067	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = 𝐺)
231	230	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹 ∘ 𝑀) = 𝐺)
232	231	fveq1d 6105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐺‘1))
233	83	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑀:(0[,]1)⟶𝐵)
234		fvco3 6185	. . . . . . . . . . . 12 ⊢ ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐹‘(𝑀‘1)))
235	233, 92, 234	sylancl 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝑀)‘1) = (𝐹‘(𝑀‘1)))
236	232, 235	eqtr3d 2646	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = (𝐹‘(𝑀‘1)))
237	228, 229, 236	3eqtrd 2648	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝑀‘1)))
238	237	mpteq2dva 4672	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1))))
239		fconstmpt 5085	. . . . . . . 8 ⊢ ((0[,]1) × {(𝐹‘(𝑀‘1))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1)))
240	238, 239	syl6eqr 2662	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
241		fovrn 6702	. . . . . . . . . 10 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
242	92, 241	mp3an2 1404	. . . . . . . . 9 ⊢ ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
243	22, 242	sylan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
244		eqidd 2611	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)))
245		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (1𝐴𝑠) → (𝐹‘𝑥) = (𝐹‘(1𝐴𝑠)))
246	243, 244, 53, 245	fmptco 6303	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))))
247		fcoconst 6307	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐵 ∧ (𝑀‘1) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
248	157, 216, 247	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
249	240, 246, 248	3eqtr4d 2654	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})))
250		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑠 = 1 → (𝑠𝐴0) = (1𝐴0))
251		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑠 = 1 → (𝑀‘𝑠) = (𝑀‘1))
252	250, 251	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑠 = 1 → ((𝑠𝐴0) = (𝑀‘𝑠) ↔ (1𝐴0) = (𝑀‘1)))
253	252	rspcv 3278	. . . . . . . 8 ⊢ (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀‘𝑠) → (1𝐴0) = (𝑀‘1)))
254	92, 90, 253	mpsyl 66	. . . . . . 7 ⊢ (𝜑 → (1𝐴0) = (𝑀‘1))
255		oveq2 6557	. . . . . . . . 9 ⊢ (𝑠 = 0 → (1𝐴𝑠) = (1𝐴0))
256		eqid 2610	. . . . . . . . 9 ⊢ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))
257		ovex 6577	. . . . . . . . 9 ⊢ (1𝐴0) ∈ V
258	255, 256, 257	fvmpt 6191	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0))
259	23, 258	ax-mp 5	. . . . . . 7 ⊢ ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0)
260		fvex 6113	. . . . . . . . 9 ⊢ (𝑀‘1) ∈ V
261	260	fvconst2 6374	. . . . . . . 8 ⊢ (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1))
262	23, 261	ax-mp 5	. . . . . . 7 ⊢ (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1)
263	254, 259, 262	3eqtr4g 2669	. . . . . 6 ⊢ (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘1)})‘0))
264	1, 19, 3, 117, 119, 62, 214, 218, 249, 263	cvmliftmoi 30519	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = ((0[,]1) × {(𝑀‘1)}))
265		fconstmpt 5085	. . . . 5 ⊢ ((0[,]1) × {(𝑀‘1)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1))
266	264, 265	syl6eq 2660	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)))
267		mpteqb 6207	. . . . 5 ⊢ (∀𝑠 ∈ (0[,]1)(1𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1)))
268		ovex 6577	. . . . . 6 ⊢ (1𝐴𝑠) ∈ V
269	268	a1i 11	. . . . 5 ⊢ (𝑠 ∈ (0[,]1) → (1𝐴𝑠) ∈ V)
270	267, 269	mprg 2910	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
271	266, 270	sylib 207	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
272	271	r19.21bi 2916	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) = (𝑀‘1))
273	8, 16, 17, 91, 212, 213, 272	isphtpy2d 22594	1 ⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 Vcvv 3173 {csn 4125 ⟨cop 4131 ∪ cuni 4372 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 0cc0 9815 1c1 9816 [,]cicc 12049 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 Conccon 21024 𝑛-Locally cnlly 21078 ×_t ctx 21173 IIcii 22486 Htpy chtpy 22574 PHtpycphtpy 22575 CovMap ccvm 30491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-cmp 21000 df-con 21025 df-lly 21079 df-nlly 21080 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-ii 22488 df-htpy 22577 df-phtpy 22578 df-phtpc 22599 df-pcon 30457 df-scon 30458 df-cvm 30492

This theorem is referenced by: cvmliftpht 30554

W3C validator