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Theorem cvmliftphtlem 31898
Description: Lemma for cvmliftpht 31899. (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 31882 . . 3 (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
87simp1d 1133 . 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 23192 . . . . . 6 (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
1312simpld 490 . . . . 5 (𝜑 → (𝐺‘0) = (𝐻‘0))
146, 13eqtrd 2814 . . . 4 (𝜑 → (𝐹𝑃) = (𝐻‘0))
151, 9, 3, 10, 5, 14cvmliftiota 31882 . . 3 (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
1615simp1d 1133 . 2 (𝜑𝑁 ∈ (II Cn 𝐶))
17 cvmliftphtlem.a . 2 (𝜑𝐴 ∈ ((II ×t II) Cn 𝐶))
18 iitop 23091 . . . . . . . . . . . . . . . 16 II ∈ Top
19 iiuni 23092 . . . . . . . . . . . . . . . 16 (0[,]1) = II
2018, 18, 19, 19txunii 21805 . . . . . . . . . . . . . . 15 ((0[,]1) × (0[,]1)) = (II ×t II)
2120, 1cnf 21458 . . . . . . . . . . . . . 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 12605 . . . . . . . . . . . . . 14 0 ∈ (0[,]1)
24 opelxpi 5392 . . . . . . . . . . . . . 14 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
2523, 24mpan2 681 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
26 fvco3 6535 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
2722, 25, 26syl2an 589 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
28 cvmliftphtlem.c . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐴) = 𝐾)
2928adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝐴) = 𝐾)
3029fveq1d 6448 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐾‘⟨𝑠, 0⟩))
3127, 30eqtr3d 2816 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 0⟩)) = (𝐾‘⟨𝑠, 0⟩))
32 df-ov 6925 . . . . . . . . . . . 12 (𝑠𝐴0) = (𝐴‘⟨𝑠, 0⟩)
3332fveq2i 6449 . . . . . . . . . . 11 (𝐹‘(𝑠𝐴0)) = (𝐹‘(𝐴‘⟨𝑠, 0⟩))
34 df-ov 6925 . . . . . . . . . . 11 (𝑠𝐾0) = (𝐾‘⟨𝑠, 0⟩)
3531, 33, 343eqtr4g 2839 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝑠𝐾0))
36 iitopon 23090 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
3736a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
384, 10phtpyhtpy 23189 . . . . . . . . . . . . 13 (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ (𝐺(II Htpy 𝐽)𝐻))
3938, 11sseldd 3822 . . . . . . . . . . . 12 (𝜑𝐾 ∈ (𝐺(II Htpy 𝐽)𝐻))
4037, 4, 10, 39htpyi 23181 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐾0) = (𝐺𝑠) ∧ (𝑠𝐾1) = (𝐻𝑠)))
4140simpld 490 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐾0) = (𝐺𝑠))
4235, 41eqtrd 2814 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝐺𝑠))
4342mpteq2dva 4979 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐺𝑠)))
44 fovrn 7081 . . . . . . . . . . 11 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
4523, 44mp3an3 1523 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
4622, 45sylan 575 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
47 eqidd 2779 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
48 cvmcn 31843 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
493, 48syl 17 . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
50 eqid 2778 . . . . . . . . . . . 12 𝐽 = 𝐽
511, 50cnf 21458 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
5249, 51syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐵 𝐽)
5352feqmptd 6509 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐵 ↦ (𝐹𝑥)))
54 fveq2 6446 . . . . . . . . 9 (𝑥 = (𝑠𝐴0) → (𝐹𝑥) = (𝐹‘(𝑠𝐴0)))
5546, 47, 53, 54fmptco 6661 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))))
5619, 50cnf 21458 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶ 𝐽)
574, 56syl 17 . . . . . . . . 9 (𝜑𝐺:(0[,]1)⟶ 𝐽)
5857feqmptd 6509 . . . . . . . 8 (𝜑𝐺 = (𝑠 ∈ (0[,]1) ↦ (𝐺𝑠)))
5943, 55, 583eqtr4d 2824 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺)
60 cvmliftphtlem.0 . . . . . . 7 (𝜑 → (0𝐴0) = 𝑃)
6137cnmptid 21873 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 𝑠) ∈ (II Cn II))
6223a1i 11 . . . . . . . . . 10 (𝜑 → 0 ∈ (0[,]1))
6337, 37, 62cnmptc 21874 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
6437, 61, 63, 17cnmpt12f 21878 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶))
651cvmlift 31880 . . . . . . . . 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 829 . . . . . . . 8 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
67 coeq2 5526 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝐹𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
6867eqeq1d 2780 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝐹𝑓) = 𝐺 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺))
69 fveq1 6445 . . . . . . . . . . . 12 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0))
70 oveq1 6929 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠𝐴0) = (0𝐴0))
71 eqid 2778 . . . . . . . . . . . . . 14 (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))
72 ovex 6954 . . . . . . . . . . . . . 14 (0𝐴0) ∈ V
7370, 71, 72fvmpt 6542 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0))
7423, 73ax-mp 5 . . . . . . . . . . . 12 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0)
7569, 74syl6eq 2830 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = (0𝐴0))
7675eqeq1d 2780 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴0) = 𝑃))
7768, 76anbi12d 624 . . . . . . . . 9 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃)))
7877riota2 6905 . . . . . . . 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 579 . . . . . . 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 702 . . . . . 6 (𝜑 → (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
812, 80syl5eq 2826 . . . . 5 (𝜑𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
8219, 1cnf 21458 . . . . . . 7 (𝑀 ∈ (II Cn 𝐶) → 𝑀:(0[,]1)⟶𝐵)
838, 82syl 17 . . . . . 6 (𝜑𝑀:(0[,]1)⟶𝐵)
8483feqmptd 6509 . . . . 5 (𝜑𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)))
8581, 84eqtr3d 2816 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)))
86 mpteqb 6560 . . . . 5 (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠)))
87 ovexd 6956 . . . . 5 (𝑠 ∈ (0[,]1) → (𝑠𝐴0) ∈ V)
8886, 87mprg 3108 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠))
8985, 88sylib 210 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠))
9089r19.21bi 3114 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴0) = (𝑀𝑠))
91 1elunit 12606 . . . . . . . . . . . . . 14 1 ∈ (0[,]1)
92 opelxpi 5392 . . . . . . . . . . . . . 14 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
9391, 92mpan2 681 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
94 fvco3 6535 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
9522, 93, 94syl2an 589 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
9629fveq1d 6448 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐾‘⟨𝑠, 1⟩))
9795, 96eqtr3d 2816 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 1⟩)) = (𝐾‘⟨𝑠, 1⟩))
98 df-ov 6925 . . . . . . . . . . . 12 (𝑠𝐴1) = (𝐴‘⟨𝑠, 1⟩)
9998fveq2i 6449 . . . . . . . . . . 11 (𝐹‘(𝑠𝐴1)) = (𝐹‘(𝐴‘⟨𝑠, 1⟩))
100 df-ov 6925 . . . . . . . . . . 11 (𝑠𝐾1) = (𝐾‘⟨𝑠, 1⟩)
10197, 99, 1003eqtr4g 2839 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝑠𝐾1))
10240simprd 491 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐾1) = (𝐻𝑠))
103101, 102eqtrd 2814 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝐻𝑠))
104103mpteq2dva 4979 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐻𝑠)))
105 fovrn 7081 . . . . . . . . . . 11 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
10691, 105mp3an3 1523 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
10722, 106sylan 575 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
108 eqidd 2779 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
109 fveq2 6446 . . . . . . . . 9 (𝑥 = (𝑠𝐴1) → (𝐹𝑥) = (𝐹‘(𝑠𝐴1)))
110107, 108, 53, 109fmptco 6661 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))))
11119, 50cnf 21458 . . . . . . . . . 10 (𝐻 ∈ (II Cn 𝐽) → 𝐻:(0[,]1)⟶ 𝐽)
11210, 111syl 17 . . . . . . . . 9 (𝜑𝐻:(0[,]1)⟶ 𝐽)
113112feqmptd 6509 . . . . . . . 8 (𝜑𝐻 = (𝑠 ∈ (0[,]1) ↦ (𝐻𝑠)))
114104, 110, 1133eqtr4d 2824 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻)
115 iiconn 23098 . . . . . . . . . . . . 13 II ∈ Conn
116115a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ Conn)
117 iinllyconn 31835 . . . . . . . . . . . . 13 II ∈ 𝑛-Locally Conn
118117a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ 𝑛-Locally Conn)
11937, 63, 61, 17cnmpt12f 21878 . . . . . . . . . . . 12 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) ∈ (II Cn 𝐶))
120 cvmtop1 31841 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
1213, 120syl 17 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ Top)
1221toptopon 21129 . . . . . . . . . . . . . 14 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
123121, 122sylib 210 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (TopOn‘𝐵))
124 ffvelrn 6621 . . . . . . . . . . . . . 14 ((𝑀:(0[,]1)⟶𝐵 ∧ 0 ∈ (0[,]1)) → (𝑀‘0) ∈ 𝐵)
12583, 23, 124sylancl 580 . . . . . . . . . . . . 13 (𝜑 → (𝑀‘0) ∈ 𝐵)
126 cnconst2 21495 . . . . . . . . . . . . 13 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘0) ∈ 𝐵) → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
12737, 123, 125, 126syl3anc 1439 . . . . . . . . . . . 12 (𝜑 → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
1284, 10, 11phtpyi 23191 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐾𝑠) = (𝐺‘0) ∧ (1𝐾𝑠) = (𝐺‘1)))
129128simpld 490 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐾𝑠) = (𝐺‘0))
130 opelxpi 5392 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
13123, 130mpan 680 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (0[,]1) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
132 fvco3 6535 . . . . . . . . . . . . . . . . . . 19 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
13322, 131, 132syl2an 589 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
13429fveq1d 6448 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐾‘⟨0, 𝑠⟩))
135133, 134eqtr3d 2816 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨0, 𝑠⟩)) = (𝐾‘⟨0, 𝑠⟩))
136 df-ov 6925 . . . . . . . . . . . . . . . . . 18 (0𝐴𝑠) = (𝐴‘⟨0, 𝑠⟩)
137136fveq2i 6449 . . . . . . . . . . . . . . . . 17 (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝐴‘⟨0, 𝑠⟩))
138 df-ov 6925 . . . . . . . . . . . . . . . . 17 (0𝐾𝑠) = (𝐾‘⟨0, 𝑠⟩)
139135, 137, 1383eqtr4g 2839 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (0𝐾𝑠))
1407simp3d 1135 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀‘0) = 𝑃)
141140adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → (𝑀‘0) = 𝑃)
142141fveq2d 6450 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐹𝑃))
1436adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑃) = (𝐺‘0))
144142, 143eqtrd 2814 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐺‘0))
145129, 139, 1443eqtr4d 2824 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝑀‘0)))
146145mpteq2dva 4979 . . . . . . . . . . . . . 14 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0))))
147 fconstmpt 5411 . . . . . . . . . . . . . 14 ((0[,]1) × {(𝐹‘(𝑀‘0))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0)))
148146, 147syl6eqr 2832 . . . . . . . . . . . . 13 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
149 fovrn 7081 . . . . . . . . . . . . . . . 16 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
15023, 149mp3an2 1522 . . . . . . . . . . . . . . 15 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
15122, 150sylan 575 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
152 eqidd 2779 . . . . . . . . . . . . . 14 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)))
153 fveq2 6446 . . . . . . . . . . . . . 14 (𝑥 = (0𝐴𝑠) → (𝐹𝑥) = (𝐹‘(0𝐴𝑠)))
154151, 152, 53, 153fmptco 6661 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))))
15552ffnd 6292 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐵)
156 fcoconst 6666 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐵 ∧ (𝑀‘0) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
157155, 125, 156syl2anc 579 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
158148, 154, 1573eqtr4d 2824 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})))
15960, 140eqtr4d 2817 . . . . . . . . . . . . 13 (𝜑 → (0𝐴0) = (𝑀‘0))
160 oveq2 6930 . . . . . . . . . . . . . . 15 (𝑠 = 0 → (0𝐴𝑠) = (0𝐴0))
161 eqid 2778 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))
162160, 161, 72fvmpt 6542 . . . . . . . . . . . . . 14 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0))
16323, 162ax-mp 5 . . . . . . . . . . . . 13 ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0)
164 fvex 6459 . . . . . . . . . . . . . . 15 (𝑀‘0) ∈ V
165164fvconst2 6741 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . 12 (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘0)})‘0))
1681, 19, 3, 116, 118, 62, 119, 127, 158, 167cvmliftmoi 31864 . . . . . . . . . . 11 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = ((0[,]1) × {(𝑀‘0)}))
169 fconstmpt 5411 . . . . . . . . . . 11 ((0[,]1) × {(𝑀‘0)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0))
170168, 169syl6eq 2830 . . . . . . . . . 10 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)))
171 mpteqb 6560 . . . . . . . . . . 11 (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0)))
172 ovexd 6956 . . . . . . . . . . 11 (𝑠 ∈ (0[,]1) → (0𝐴𝑠) ∈ V)
173171, 172mprg 3108 . . . . . . . . . 10 ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
174170, 173sylib 210 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
175 oveq2 6930 . . . . . . . . . . 11 (𝑠 = 1 → (0𝐴𝑠) = (0𝐴1))
176175eqeq1d 2780 . . . . . . . . . 10 (𝑠 = 1 → ((0𝐴𝑠) = (𝑀‘0) ↔ (0𝐴1) = (𝑀‘0)))
177176rspcv 3507 . . . . . . . . 9 (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0) → (0𝐴1) = (𝑀‘0)))
17891, 174, 177mpsyl 68 . . . . . . . 8 (𝜑 → (0𝐴1) = (𝑀‘0))
179178, 140eqtrd 2814 . . . . . . 7 (𝜑 → (0𝐴1) = 𝑃)
18091a1i 11 . . . . . . . . . 10 (𝜑 → 1 ∈ (0[,]1))
18137, 37, 180cnmptc 21874 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 1) ∈ (II Cn II))
18237, 61, 181, 17cnmpt12f 21878 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶))
1831cvmlift 31880 . . . . . . . . 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 829 . . . . . . . 8 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
185 coeq2 5526 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝐹𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
186185eqeq1d 2780 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝐹𝑓) = 𝐻 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻))
187 fveq1 6445 . . . . . . . . . . . 12 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0))
188 oveq1 6929 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠𝐴1) = (0𝐴1))
189 eqid 2778 . . . . . . . . . . . . . 14 (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))
190 ovex 6954 . . . . . . . . . . . . . 14 (0𝐴1) ∈ V
191188, 189, 190fvmpt 6542 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1))
19223, 191ax-mp 5 . . . . . . . . . . . 12 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1)
193187, 192syl6eq 2830 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = (0𝐴1))
194193eqeq1d 2780 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴1) = 𝑃))
195186, 194anbi12d 624 . . . . . . . . 9 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃)))
196195riota2 6905 . . . . . . . 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 579 . . . . . . 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 702 . . . . . 6 (𝜑 → (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
1999, 198syl5eq 2826 . . . . 5 (𝜑𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
20019, 1cnf 21458 . . . . . . 7 (𝑁 ∈ (II Cn 𝐶) → 𝑁:(0[,]1)⟶𝐵)
20116, 200syl 17 . . . . . 6 (𝜑𝑁:(0[,]1)⟶𝐵)
202201feqmptd 6509 . . . . 5 (𝜑𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)))
203199, 202eqtr3d 2816 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)))
204 mpteqb 6560 . . . . 5 (∀𝑠 ∈ (0[,]1)(𝑠𝐴1) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠)))
205 ovexd 6956 . . . . 5 (𝑠 ∈ (0[,]1) → (𝑠𝐴1) ∈ V)
206204, 205mprg 3108 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠))
207203, 206sylib 210 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠))
208207r19.21bi 3114 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴1) = (𝑁𝑠))
209174r19.21bi 3114 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐴𝑠) = (𝑀‘0))
21037, 181, 61, 17cnmpt12f 21878 . . . . . 6 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) ∈ (II Cn 𝐶))
211 ffvelrn 6621 . . . . . . . 8 ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → (𝑀‘1) ∈ 𝐵)
21283, 91, 211sylancl 580 . . . . . . 7 (𝜑 → (𝑀‘1) ∈ 𝐵)
213 cnconst2 21495 . . . . . . 7 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘1) ∈ 𝐵) → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
21437, 123, 212, 213syl3anc 1439 . . . . . 6 (𝜑 → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
215 opelxpi 5392 . . . . . . . . . . . . . 14 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
21691, 215mpan 680 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
217 fvco3 6535 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
21822, 216, 217syl2an 589 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
21929fveq1d 6448 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐾‘⟨1, 𝑠⟩))
220218, 219eqtr3d 2816 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨1, 𝑠⟩)) = (𝐾‘⟨1, 𝑠⟩))
221 df-ov 6925 . . . . . . . . . . . 12 (1𝐴𝑠) = (𝐴‘⟨1, 𝑠⟩)
222221fveq2i 6449 . . . . . . . . . . 11 (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝐴‘⟨1, 𝑠⟩))
223 df-ov 6925 . . . . . . . . . . 11 (1𝐾𝑠) = (𝐾‘⟨1, 𝑠⟩)
224220, 222, 2233eqtr4g 2839 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (1𝐾𝑠))
225128simprd 491 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐾𝑠) = (𝐺‘1))
2267simp2d 1134 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑀) = 𝐺)
227226adantr 474 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑀) = 𝐺)
228227fveq1d 6448 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐺‘1))
22983adantr 474 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → 𝑀:(0[,]1)⟶𝐵)
230 fvco3 6535 . . . . . . . . . . . 12 ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐹‘(𝑀‘1)))
231229, 91, 230sylancl 580 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐹‘(𝑀‘1)))
232228, 231eqtr3d 2816 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐺‘1) = (𝐹‘(𝑀‘1)))
233224, 225, 2323eqtrd 2818 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝑀‘1)))
234233mpteq2dva 4979 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1))))
235 fconstmpt 5411 . . . . . . . 8 ((0[,]1) × {(𝐹‘(𝑀‘1))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1)))
236234, 235syl6eqr 2832 . . . . . . 7 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
237 fovrn 7081 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
23891, 237mp3an2 1522 . . . . . . . . 9 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
23922, 238sylan 575 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
240 eqidd 2779 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)))
241 fveq2 6446 . . . . . . . 8 (𝑥 = (1𝐴𝑠) → (𝐹𝑥) = (𝐹‘(1𝐴𝑠)))
242239, 240, 53, 241fmptco 6661 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))))
243 fcoconst 6666 . . . . . . . 8 ((𝐹 Fn 𝐵 ∧ (𝑀‘1) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
244155, 212, 243syl2anc 579 . . . . . . 7 (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
245236, 242, 2443eqtr4d 2824 . . . . . 6 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})))
246 oveq1 6929 . . . . . . . . . 10 (𝑠 = 1 → (𝑠𝐴0) = (1𝐴0))
247 fveq2 6446 . . . . . . . . . 10 (𝑠 = 1 → (𝑀𝑠) = (𝑀‘1))
248246, 247eqeq12d 2793 . . . . . . . . 9 (𝑠 = 1 → ((𝑠𝐴0) = (𝑀𝑠) ↔ (1𝐴0) = (𝑀‘1)))
249248rspcv 3507 . . . . . . . 8 (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠) → (1𝐴0) = (𝑀‘1)))
25091, 89, 249mpsyl 68 . . . . . . 7 (𝜑 → (1𝐴0) = (𝑀‘1))
251 oveq2 6930 . . . . . . . . 9 (𝑠 = 0 → (1𝐴𝑠) = (1𝐴0))
252 eqid 2778 . . . . . . . . 9 (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))
253 ovex 6954 . . . . . . . . 9 (1𝐴0) ∈ V
254251, 252, 253fvmpt 6542 . . . . . . . 8 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0))
25523, 254ax-mp 5 . . . . . . 7 ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0)
256 fvex 6459 . . . . . . . . 9 (𝑀‘1) ∈ V
257256fvconst2 6741 . . . . . . . 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 2839 . . . . . 6 (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘1)})‘0))
2601, 19, 3, 116, 118, 62, 210, 214, 245, 259cvmliftmoi 31864 . . . . 5 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = ((0[,]1) × {(𝑀‘1)}))
261 fconstmpt 5411 . . . . 5 ((0[,]1) × {(𝑀‘1)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1))
262260, 261syl6eq 2830 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)))
263 mpteqb 6560 . . . . 5 (∀𝑠 ∈ (0[,]1)(1𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1)))
264 ovexd 6956 . . . . 5 (𝑠 ∈ (0[,]1) → (1𝐴𝑠) ∈ V)
265263, 264mprg 3108 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
266262, 265sylib 210 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
267266r19.21bi 3114 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐴𝑠) = (𝑀‘1))
2688, 16, 17, 90, 208, 209, 267isphtpy2d 23194 1 (𝜑𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  ∃!wreu 3092  Vcvv 3398  {csn 4398  cop 4404   cuni 4671  cmpt 4965   × cxp 5353  ccom 5359   Fn wfn 6130  wf 6131  cfv 6135  crio 6882  (class class class)co 6922  0cc0 10272  1c1 10273  [,]cicc 12490  Topctop 21105  TopOnctopon 21122   Cn ccn 21436  Conncconn 21623  𝑛-Locally cnlly 21677   ×t ctx 21772  IIcii 23086   Htpy chtpy 23174  PHtpycphtpy 23175   CovMap ccvm 31836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351  ax-mulf 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-ec 8028  df-map 8142  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-fi 8605  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-ioo 12491  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-fl 12912  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-starv 16353  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-unif 16361  df-hom 16362  df-cco 16363  df-rest 16469  df-topn 16470  df-0g 16488  df-gsum 16489  df-topgen 16490  df-pt 16491  df-prds 16494  df-xrs 16548  df-qtop 16553  df-imas 16554  df-xps 16556  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-mulg 17928  df-cntz 18133  df-cmn 18581  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-cnfld 20143  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-cn 21439  df-cnp 21440  df-cmp 21599  df-conn 21624  df-lly 21678  df-nlly 21679  df-tx 21774  df-hmeo 21967  df-xms 22533  df-ms 22534  df-tms 22535  df-ii 23088  df-htpy 23177  df-phtpy 23178  df-phtpc 23199  df-pconn 31802  df-sconn 31803  df-cvm 31837
This theorem is referenced by:  cvmliftpht  31899
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