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Theorem cvmliftphtlem 32461
Description: Lemma for cvmliftpht 32462. (Contributed by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
cvmliftpht.b 𝐵 = 𝐶
cvmliftpht.m 𝑀 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
cvmliftpht.n 𝑁 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
cvmliftpht.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftpht.p (𝜑𝑃𝐵)
cvmliftpht.e (𝜑 → (𝐹𝑃) = (𝐺‘0))
cvmliftphtlem.g (𝜑𝐺 ∈ (II Cn 𝐽))
cvmliftphtlem.h (𝜑𝐻 ∈ (II Cn 𝐽))
cvmliftphtlem.k (𝜑𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))
cvmliftphtlem.a (𝜑𝐴 ∈ ((II ×t II) Cn 𝐶))
cvmliftphtlem.c (𝜑 → (𝐹𝐴) = 𝐾)
cvmliftphtlem.0 (𝜑 → (0𝐴0) = 𝑃)
Assertion
Ref Expression
cvmliftphtlem (𝜑𝐴 ∈ (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.
StepHypRef Expression
1 cvmliftpht.b . . . 4 𝐵 = 𝐶
2 cvmliftpht.m . . . 4 𝑀 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
3 cvmliftpht.f . . . 4 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmliftphtlem.g . . . 4 (𝜑𝐺 ∈ (II Cn 𝐽))
5 cvmliftpht.p . . . 4 (𝜑𝑃𝐵)
6 cvmliftpht.e . . . 4 (𝜑 → (𝐹𝑃) = (𝐺‘0))
71, 2, 3, 4, 5, 6cvmliftiota 32445 . . 3 (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
87simp1d 1134 . 2 (𝜑𝑀 ∈ (II Cn 𝐶))
9 cvmliftpht.n . . . 4 𝑁 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
10 cvmliftphtlem.h . . . 4 (𝜑𝐻 ∈ (II Cn 𝐽))
11 cvmliftphtlem.k . . . . . . 7 (𝜑𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))
124, 10, 11phtpy01 23516 . . . . . 6 (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
1312simpld 495 . . . . 5 (𝜑 → (𝐺‘0) = (𝐻‘0))
146, 13eqtrd 2853 . . . 4 (𝜑 → (𝐹𝑃) = (𝐻‘0))
151, 9, 3, 10, 5, 14cvmliftiota 32445 . . 3 (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
1615simp1d 1134 . 2 (𝜑𝑁 ∈ (II Cn 𝐶))
17 cvmliftphtlem.a . 2 (𝜑𝐴 ∈ ((II ×t II) Cn 𝐶))
18 iitop 23415 . . . . . . . . . . . . . . . 16 II ∈ Top
19 iiuni 23416 . . . . . . . . . . . . . . . 16 (0[,]1) = II
2018, 18, 19, 19txunii 22129 . . . . . . . . . . . . . . 15 ((0[,]1) × (0[,]1)) = (II ×t II)
2120, 1cnf 21782 . . . . . . . . . . . . . 14 (𝐴 ∈ ((II ×t II) Cn 𝐶) → 𝐴:((0[,]1) × (0[,]1))⟶𝐵)
2217, 21syl 17 . . . . . . . . . . . . 13 (𝜑𝐴:((0[,]1) × (0[,]1))⟶𝐵)
23 0elunit 12843 . . . . . . . . . . . . . 14 0 ∈ (0[,]1)
24 opelxpi 5585 . . . . . . . . . . . . . 14 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
2523, 24mpan2 687 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
26 fvco3 6753 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
2722, 25, 26syl2an 595 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
28 cvmliftphtlem.c . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐴) = 𝐾)
2928adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝐴) = 𝐾)
3029fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐾‘⟨𝑠, 0⟩))
3127, 30eqtr3d 2855 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 0⟩)) = (𝐾‘⟨𝑠, 0⟩))
32 df-ov 7148 . . . . . . . . . . . 12 (𝑠𝐴0) = (𝐴‘⟨𝑠, 0⟩)
3332fveq2i 6666 . . . . . . . . . . 11 (𝐹‘(𝑠𝐴0)) = (𝐹‘(𝐴‘⟨𝑠, 0⟩))
34 df-ov 7148 . . . . . . . . . . 11 (𝑠𝐾0) = (𝐾‘⟨𝑠, 0⟩)
3531, 33, 343eqtr4g 2878 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝑠𝐾0))
36 iitopon 23414 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
3736a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
384, 10phtpyhtpy 23513 . . . . . . . . . . . . 13 (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ (𝐺(II Htpy 𝐽)𝐻))
3938, 11sseldd 3965 . . . . . . . . . . . 12 (𝜑𝐾 ∈ (𝐺(II Htpy 𝐽)𝐻))
4037, 4, 10, 39htpyi 23505 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐾0) = (𝐺𝑠) ∧ (𝑠𝐾1) = (𝐻𝑠)))
4140simpld 495 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐾0) = (𝐺𝑠))
4235, 41eqtrd 2853 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝐺𝑠))
4342mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐺𝑠)))
44 fovrn 7307 . . . . . . . . . . 11 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
4523, 44mp3an3 1441 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
4622, 45sylan 580 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
47 eqidd 2819 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
48 cvmcn 32406 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
493, 48syl 17 . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
50 eqid 2818 . . . . . . . . . . . 12 𝐽 = 𝐽
511, 50cnf 21782 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
5249, 51syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐵 𝐽)
5352feqmptd 6726 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐵 ↦ (𝐹𝑥)))
54 fveq2 6663 . . . . . . . . 9 (𝑥 = (𝑠𝐴0) → (𝐹𝑥) = (𝐹‘(𝑠𝐴0)))
5546, 47, 53, 54fmptco 6883 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))))
5619, 50cnf 21782 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶ 𝐽)
574, 56syl 17 . . . . . . . . 9 (𝜑𝐺:(0[,]1)⟶ 𝐽)
5857feqmptd 6726 . . . . . . . 8 (𝜑𝐺 = (𝑠 ∈ (0[,]1) ↦ (𝐺𝑠)))
5943, 55, 583eqtr4d 2863 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺)
60 cvmliftphtlem.0 . . . . . . 7 (𝜑 → (0𝐴0) = 𝑃)
6137cnmptid 22197 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 𝑠) ∈ (II Cn II))
6223a1i 11 . . . . . . . . . 10 (𝜑 → 0 ∈ (0[,]1))
6337, 37, 62cnmptc 22198 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
6437, 61, 63, 17cnmpt12f 22202 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶))
651cvmlift 32443 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃𝐵 ∧ (𝐹𝑃) = (𝐺‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
663, 4, 5, 6, 65syl22anc 834 . . . . . . . 8 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
67 coeq2 5722 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝐹𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
6867eqeq1d 2820 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝐹𝑓) = 𝐺 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺))
69 fveq1 6662 . . . . . . . . . . . 12 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0))
70 oveq1 7152 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠𝐴0) = (0𝐴0))
71 eqid 2818 . . . . . . . . . . . . . 14 (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))
72 ovex 7178 . . . . . . . . . . . . . 14 (0𝐴0) ∈ V
7370, 71, 72fvmpt 6761 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0))
7423, 73ax-mp 5 . . . . . . . . . . . 12 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0)
7569, 74syl6eq 2869 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = (0𝐴0))
7675eqeq1d 2820 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴0) = 𝑃))
7768, 76anbi12d 630 . . . . . . . . 9 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃)))
7877riota2 7128 . . . . . . . 8 (((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶) ∧ ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃) ↔ (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
7964, 66, 78syl2anc 584 . . . . . . 7 (𝜑 → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃) ↔ (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
8059, 60, 79mpbi2and 708 . . . . . 6 (𝜑 → (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
812, 80syl5eq 2865 . . . . 5 (𝜑𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
8219, 1cnf 21782 . . . . . . 7 (𝑀 ∈ (II Cn 𝐶) → 𝑀:(0[,]1)⟶𝐵)
838, 82syl 17 . . . . . 6 (𝜑𝑀:(0[,]1)⟶𝐵)
8483feqmptd 6726 . . . . 5 (𝜑𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)))
8581, 84eqtr3d 2855 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)))
86 mpteqb 6779 . . . . 5 (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠)))
87 ovexd 7180 . . . . 5 (𝑠 ∈ (0[,]1) → (𝑠𝐴0) ∈ V)
8886, 87mprg 3149 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠))
8985, 88sylib 219 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠))
9089r19.21bi 3205 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴0) = (𝑀𝑠))
91 1elunit 12844 . . . . . . . . . . . . . 14 1 ∈ (0[,]1)
92 opelxpi 5585 . . . . . . . . . . . . . 14 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
9391, 92mpan2 687 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
94 fvco3 6753 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
9522, 93, 94syl2an 595 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
9629fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐾‘⟨𝑠, 1⟩))
9795, 96eqtr3d 2855 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 1⟩)) = (𝐾‘⟨𝑠, 1⟩))
98 df-ov 7148 . . . . . . . . . . . 12 (𝑠𝐴1) = (𝐴‘⟨𝑠, 1⟩)
9998fveq2i 6666 . . . . . . . . . . 11 (𝐹‘(𝑠𝐴1)) = (𝐹‘(𝐴‘⟨𝑠, 1⟩))
100 df-ov 7148 . . . . . . . . . . 11 (𝑠𝐾1) = (𝐾‘⟨𝑠, 1⟩)
10197, 99, 1003eqtr4g 2878 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝑠𝐾1))
10240simprd 496 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐾1) = (𝐻𝑠))
103101, 102eqtrd 2853 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝐻𝑠))
104103mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐻𝑠)))
105 fovrn 7307 . . . . . . . . . . 11 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
10691, 105mp3an3 1441 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
10722, 106sylan 580 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
108 eqidd 2819 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
109 fveq2 6663 . . . . . . . . 9 (𝑥 = (𝑠𝐴1) → (𝐹𝑥) = (𝐹‘(𝑠𝐴1)))
110107, 108, 53, 109fmptco 6883 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))))
11119, 50cnf 21782 . . . . . . . . . 10 (𝐻 ∈ (II Cn 𝐽) → 𝐻:(0[,]1)⟶ 𝐽)
11210, 111syl 17 . . . . . . . . 9 (𝜑𝐻:(0[,]1)⟶ 𝐽)
113112feqmptd 6726 . . . . . . . 8 (𝜑𝐻 = (𝑠 ∈ (0[,]1) ↦ (𝐻𝑠)))
114104, 110, 1133eqtr4d 2863 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻)
115 iiconn 23422 . . . . . . . . . . . . 13 II ∈ Conn
116115a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ Conn)
117 iinllyconn 32398 . . . . . . . . . . . . 13 II ∈ 𝑛-Locally Conn
118117a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ 𝑛-Locally Conn)
11937, 63, 61, 17cnmpt12f 22202 . . . . . . . . . . . 12 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) ∈ (II Cn 𝐶))
120 cvmtop1 32404 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
1213, 120syl 17 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ Top)
1221toptopon 21453 . . . . . . . . . . . . . 14 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
123121, 122sylib 219 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (TopOn‘𝐵))
124 ffvelrn 6841 . . . . . . . . . . . . . 14 ((𝑀:(0[,]1)⟶𝐵 ∧ 0 ∈ (0[,]1)) → (𝑀‘0) ∈ 𝐵)
12583, 23, 124sylancl 586 . . . . . . . . . . . . 13 (𝜑 → (𝑀‘0) ∈ 𝐵)
126 cnconst2 21819 . . . . . . . . . . . . 13 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘0) ∈ 𝐵) → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
12737, 123, 125, 126syl3anc 1363 . . . . . . . . . . . 12 (𝜑 → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
1284, 10, 11phtpyi 23515 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐾𝑠) = (𝐺‘0) ∧ (1𝐾𝑠) = (𝐺‘1)))
129128simpld 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐾𝑠) = (𝐺‘0))
130 opelxpi 5585 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
13123, 130mpan 686 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (0[,]1) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
132 fvco3 6753 . . . . . . . . . . . . . . . . . . 19 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
13322, 131, 132syl2an 595 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
13429fveq1d 6665 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐾‘⟨0, 𝑠⟩))
135133, 134eqtr3d 2855 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨0, 𝑠⟩)) = (𝐾‘⟨0, 𝑠⟩))
136 df-ov 7148 . . . . . . . . . . . . . . . . . 18 (0𝐴𝑠) = (𝐴‘⟨0, 𝑠⟩)
137136fveq2i 6666 . . . . . . . . . . . . . . . . 17 (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝐴‘⟨0, 𝑠⟩))
138 df-ov 7148 . . . . . . . . . . . . . . . . 17 (0𝐾𝑠) = (𝐾‘⟨0, 𝑠⟩)
139135, 137, 1383eqtr4g 2878 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (0𝐾𝑠))
1407simp3d 1136 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀‘0) = 𝑃)
141140adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → (𝑀‘0) = 𝑃)
142141fveq2d 6667 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐹𝑃))
1436adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑃) = (𝐺‘0))
144142, 143eqtrd 2853 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐺‘0))
145129, 139, 1443eqtr4d 2863 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝑀‘0)))
146145mpteq2dva 5152 . . . . . . . . . . . . . 14 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0))))
147 fconstmpt 5607 . . . . . . . . . . . . . 14 ((0[,]1) × {(𝐹‘(𝑀‘0))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0)))
148146, 147syl6eqr 2871 . . . . . . . . . . . . 13 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
149 fovrn 7307 . . . . . . . . . . . . . . . 16 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
15023, 149mp3an2 1440 . . . . . . . . . . . . . . 15 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
15122, 150sylan 580 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
152 eqidd 2819 . . . . . . . . . . . . . 14 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)))
153 fveq2 6663 . . . . . . . . . . . . . 14 (𝑥 = (0𝐴𝑠) → (𝐹𝑥) = (𝐹‘(0𝐴𝑠)))
154151, 152, 53, 153fmptco 6883 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))))
15552ffnd 6508 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐵)
156 fcoconst 6888 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐵 ∧ (𝑀‘0) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
157155, 125, 156syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
158148, 154, 1573eqtr4d 2863 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})))
15960, 140eqtr4d 2856 . . . . . . . . . . . . 13 (𝜑 → (0𝐴0) = (𝑀‘0))
160 oveq2 7153 . . . . . . . . . . . . . . 15 (𝑠 = 0 → (0𝐴𝑠) = (0𝐴0))
161 eqid 2818 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))
162160, 161, 72fvmpt 6761 . . . . . . . . . . . . . 14 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0))
16323, 162ax-mp 5 . . . . . . . . . . . . 13 ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0)
164 fvex 6676 . . . . . . . . . . . . . . 15 (𝑀‘0) ∈ V
165164fvconst2 6958 . . . . . . . . . . . . . 14 (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0))
16623, 165ax-mp 5 . . . . . . . . . . . . 13 (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0)
167159, 163, 1663eqtr4g 2878 . . . . . . . . . . . 12 (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘0)})‘0))
1681, 19, 3, 116, 118, 62, 119, 127, 158, 167cvmliftmoi 32427 . . . . . . . . . . 11 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = ((0[,]1) × {(𝑀‘0)}))
169 fconstmpt 5607 . . . . . . . . . . 11 ((0[,]1) × {(𝑀‘0)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0))
170168, 169syl6eq 2869 . . . . . . . . . 10 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)))
171 mpteqb 6779 . . . . . . . . . . 11 (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0)))
172 ovexd 7180 . . . . . . . . . . 11 (𝑠 ∈ (0[,]1) → (0𝐴𝑠) ∈ V)
173171, 172mprg 3149 . . . . . . . . . 10 ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
174170, 173sylib 219 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
175 oveq2 7153 . . . . . . . . . . 11 (𝑠 = 1 → (0𝐴𝑠) = (0𝐴1))
176175eqeq1d 2820 . . . . . . . . . 10 (𝑠 = 1 → ((0𝐴𝑠) = (𝑀‘0) ↔ (0𝐴1) = (𝑀‘0)))
177176rspcv 3615 . . . . . . . . 9 (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0) → (0𝐴1) = (𝑀‘0)))
17891, 174, 177mpsyl 68 . . . . . . . 8 (𝜑 → (0𝐴1) = (𝑀‘0))
179178, 140eqtrd 2853 . . . . . . 7 (𝜑 → (0𝐴1) = 𝑃)
18091a1i 11 . . . . . . . . . 10 (𝜑 → 1 ∈ (0[,]1))
18137, 37, 180cnmptc 22198 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 1) ∈ (II Cn II))
18237, 61, 181, 17cnmpt12f 22202 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶))
1831cvmlift 32443 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽)) ∧ (𝑃𝐵 ∧ (𝐹𝑃) = (𝐻‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
1843, 10, 5, 14, 183syl22anc 834 . . . . . . . 8 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
185 coeq2 5722 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝐹𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
186185eqeq1d 2820 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝐹𝑓) = 𝐻 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻))
187 fveq1 6662 . . . . . . . . . . . 12 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0))
188 oveq1 7152 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠𝐴1) = (0𝐴1))
189 eqid 2818 . . . . . . . . . . . . . 14 (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))
190 ovex 7178 . . . . . . . . . . . . . 14 (0𝐴1) ∈ V
191188, 189, 190fvmpt 6761 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1))
19223, 191ax-mp 5 . . . . . . . . . . . 12 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1)
193187, 192syl6eq 2869 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = (0𝐴1))
194193eqeq1d 2820 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴1) = 𝑃))
195186, 194anbi12d 630 . . . . . . . . 9 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃)))
196195riota2 7128 . . . . . . . 8 (((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶) ∧ ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃) ↔ (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
197182, 184, 196syl2anc 584 . . . . . . 7 (𝜑 → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃) ↔ (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
198114, 179, 197mpbi2and 708 . . . . . 6 (𝜑 → (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
1999, 198syl5eq 2865 . . . . 5 (𝜑𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
20019, 1cnf 21782 . . . . . . 7 (𝑁 ∈ (II Cn 𝐶) → 𝑁:(0[,]1)⟶𝐵)
20116, 200syl 17 . . . . . 6 (𝜑𝑁:(0[,]1)⟶𝐵)
202201feqmptd 6726 . . . . 5 (𝜑𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)))
203199, 202eqtr3d 2855 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)))
204 mpteqb 6779 . . . . 5 (∀𝑠 ∈ (0[,]1)(𝑠𝐴1) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠)))
205 ovexd 7180 . . . . 5 (𝑠 ∈ (0[,]1) → (𝑠𝐴1) ∈ V)
206204, 205mprg 3149 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠))
207203, 206sylib 219 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠))
208207r19.21bi 3205 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴1) = (𝑁𝑠))
209174r19.21bi 3205 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐴𝑠) = (𝑀‘0))
21037, 181, 61, 17cnmpt12f 22202 . . . . . 6 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) ∈ (II Cn 𝐶))
211 ffvelrn 6841 . . . . . . . 8 ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → (𝑀‘1) ∈ 𝐵)
21283, 91, 211sylancl 586 . . . . . . 7 (𝜑 → (𝑀‘1) ∈ 𝐵)
213 cnconst2 21819 . . . . . . 7 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘1) ∈ 𝐵) → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
21437, 123, 212, 213syl3anc 1363 . . . . . 6 (𝜑 → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
215 opelxpi 5585 . . . . . . . . . . . . . 14 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
21691, 215mpan 686 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
217 fvco3 6753 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
21822, 216, 217syl2an 595 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
21929fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐾‘⟨1, 𝑠⟩))
220218, 219eqtr3d 2855 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨1, 𝑠⟩)) = (𝐾‘⟨1, 𝑠⟩))
221 df-ov 7148 . . . . . . . . . . . 12 (1𝐴𝑠) = (𝐴‘⟨1, 𝑠⟩)
222221fveq2i 6666 . . . . . . . . . . 11 (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝐴‘⟨1, 𝑠⟩))
223 df-ov 7148 . . . . . . . . . . 11 (1𝐾𝑠) = (𝐾‘⟨1, 𝑠⟩)
224220, 222, 2233eqtr4g 2878 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (1𝐾𝑠))
225128simprd 496 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐾𝑠) = (𝐺‘1))
2267simp2d 1135 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑀) = 𝐺)
227226adantr 481 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑀) = 𝐺)
228227fveq1d 6665 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐺‘1))
22983adantr 481 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → 𝑀:(0[,]1)⟶𝐵)
230 fvco3 6753 . . . . . . . . . . . 12 ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐹‘(𝑀‘1)))
231229, 91, 230sylancl 586 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐹‘(𝑀‘1)))
232228, 231eqtr3d 2855 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐺‘1) = (𝐹‘(𝑀‘1)))
233224, 225, 2323eqtrd 2857 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝑀‘1)))
234233mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1))))
235 fconstmpt 5607 . . . . . . . 8 ((0[,]1) × {(𝐹‘(𝑀‘1))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1)))
236234, 235syl6eqr 2871 . . . . . . 7 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
237 fovrn 7307 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
23891, 237mp3an2 1440 . . . . . . . . 9 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
23922, 238sylan 580 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
240 eqidd 2819 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)))
241 fveq2 6663 . . . . . . . 8 (𝑥 = (1𝐴𝑠) → (𝐹𝑥) = (𝐹‘(1𝐴𝑠)))
242239, 240, 53, 241fmptco 6883 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))))
243 fcoconst 6888 . . . . . . . 8 ((𝐹 Fn 𝐵 ∧ (𝑀‘1) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
244155, 212, 243syl2anc 584 . . . . . . 7 (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
245236, 242, 2443eqtr4d 2863 . . . . . 6 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})))
246 oveq1 7152 . . . . . . . . . 10 (𝑠 = 1 → (𝑠𝐴0) = (1𝐴0))
247 fveq2 6663 . . . . . . . . . 10 (𝑠 = 1 → (𝑀𝑠) = (𝑀‘1))
248246, 247eqeq12d 2834 . . . . . . . . 9 (𝑠 = 1 → ((𝑠𝐴0) = (𝑀𝑠) ↔ (1𝐴0) = (𝑀‘1)))
249248rspcv 3615 . . . . . . . 8 (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠) → (1𝐴0) = (𝑀‘1)))
25091, 89, 249mpsyl 68 . . . . . . 7 (𝜑 → (1𝐴0) = (𝑀‘1))
251 oveq2 7153 . . . . . . . . 9 (𝑠 = 0 → (1𝐴𝑠) = (1𝐴0))
252 eqid 2818 . . . . . . . . 9 (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))
253 ovex 7178 . . . . . . . . 9 (1𝐴0) ∈ V
254251, 252, 253fvmpt 6761 . . . . . . . 8 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0))
25523, 254ax-mp 5 . . . . . . 7 ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0)
256 fvex 6676 . . . . . . . . 9 (𝑀‘1) ∈ V
257256fvconst2 6958 . . . . . . . 8 (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1))
25823, 257ax-mp 5 . . . . . . 7 (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1)
259250, 255, 2583eqtr4g 2878 . . . . . 6 (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘1)})‘0))
2601, 19, 3, 116, 118, 62, 210, 214, 245, 259cvmliftmoi 32427 . . . . 5 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = ((0[,]1) × {(𝑀‘1)}))
261 fconstmpt 5607 . . . . 5 ((0[,]1) × {(𝑀‘1)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1))
262260, 261syl6eq 2869 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)))
263 mpteqb 6779 . . . . 5 (∀𝑠 ∈ (0[,]1)(1𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1)))
264 ovexd 7180 . . . . 5 (𝑠 ∈ (0[,]1) → (1𝐴𝑠) ∈ V)
265263, 264mprg 3149 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
266262, 265sylib 219 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
267266r19.21bi 3205 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐴𝑠) = (𝑀‘1))
2688, 16, 17, 90, 208, 209, 267isphtpy2d 23518 1 (𝜑𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  ∃!wreu 3137  Vcvv 3492  {csn 4557  cop 4563   cuni 4830  cmpt 5137   × cxp 5546  ccom 5552   Fn wfn 6343  wf 6344  cfv 6348  crio 7102  (class class class)co 7145  0cc0 10525  1c1 10526  [,]cicc 12729  Topctop 21429  TopOnctopon 21446   Cn ccn 21760  Conncconn 21947  𝑛-Locally cnlly 22001   ×t ctx 22096  IIcii 23410   Htpy chtpy 23498  PHtpycphtpy 23499   CovMap ccvm 32399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-ec 8280  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cn 21763  df-cnp 21764  df-cmp 21923  df-conn 21948  df-lly 22002  df-nlly 22003  df-tx 22098  df-hmeo 22291  df-xms 22857  df-ms 22858  df-tms 22859  df-ii 23412  df-htpy 23501  df-phtpy 23502  df-phtpc 23523  df-pconn 32365  df-sconn 32366  df-cvm 32400
This theorem is referenced by:  cvmliftpht  32462
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