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Theorem cvmliftphtlem 31004
Description: Lemma for cvmliftpht 31005. (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 30988 . . 3 (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
87simp1d 1071 . 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 22692 . . . . . 6 (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
1312simpld 475 . . . . 5 (𝜑 → (𝐺‘0) = (𝐻‘0))
146, 13eqtrd 2655 . . . 4 (𝜑 → (𝐹𝑃) = (𝐻‘0))
151, 9, 3, 10, 5, 14cvmliftiota 30988 . . 3 (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
1615simp1d 1071 . 2 (𝜑𝑁 ∈ (II Cn 𝐶))
17 cvmliftphtlem.a . 2 (𝜑𝐴 ∈ ((II ×t II) Cn 𝐶))
18 iitop 22591 . . . . . . . . . . . . . . . 16 II ∈ Top
19 iiuni 22592 . . . . . . . . . . . . . . . 16 (0[,]1) = II
2018, 18, 19, 19txunii 21306 . . . . . . . . . . . . . . 15 ((0[,]1) × (0[,]1)) = (II ×t II)
2120, 1cnf 20960 . . . . . . . . . . . . . 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 12232 . . . . . . . . . . . . . 14 0 ∈ (0[,]1)
24 opelxpi 5108 . . . . . . . . . . . . . 14 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
2523, 24mpan2 706 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1)))
26 fvco3 6232 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 0⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐹‘(𝐴‘⟨𝑠, 0⟩)))
2722, 25, 26syl2an 494 . . . . . . . . . . . 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 6150 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 0⟩) = (𝐾‘⟨𝑠, 0⟩))
3127, 30eqtr3d 2657 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 0⟩)) = (𝐾‘⟨𝑠, 0⟩))
32 df-ov 6607 . . . . . . . . . . . 12 (𝑠𝐴0) = (𝐴‘⟨𝑠, 0⟩)
3332fveq2i 6151 . . . . . . . . . . 11 (𝐹‘(𝑠𝐴0)) = (𝐹‘(𝐴‘⟨𝑠, 0⟩))
34 df-ov 6607 . . . . . . . . . . 11 (𝑠𝐾0) = (𝐾‘⟨𝑠, 0⟩)
3531, 33, 343eqtr4g 2680 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝑠𝐾0))
36 iitopon 22590 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
3736a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
384, 10phtpyhtpy 22689 . . . . . . . . . . . . 13 (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ (𝐺(II Htpy 𝐽)𝐻))
3938, 11sseldd 3584 . . . . . . . . . . . 12 (𝜑𝐾 ∈ (𝐺(II Htpy 𝐽)𝐻))
4037, 4, 10, 39htpyi 22681 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐾0) = (𝐺𝑠) ∧ (𝑠𝐾1) = (𝐻𝑠)))
4140simpld 475 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐾0) = (𝐺𝑠))
4235, 41eqtrd 2655 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴0)) = (𝐺𝑠))
4342mpteq2dva 4704 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐺𝑠)))
44 fovrn 6757 . . . . . . . . . . 11 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
4523, 44mp3an3 1410 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
4622, 45sylan 488 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴0) ∈ 𝐵)
47 eqidd 2622 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
48 cvmcn 30949 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
493, 48syl 17 . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
50 eqid 2621 . . . . . . . . . . . 12 𝐽 = 𝐽
511, 50cnf 20960 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
5249, 51syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐵 𝐽)
5352feqmptd 6206 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐵 ↦ (𝐹𝑥)))
54 fveq2 6148 . . . . . . . . 9 (𝑥 = (𝑠𝐴0) → (𝐹𝑥) = (𝐹‘(𝑠𝐴0)))
5546, 47, 53, 54fmptco 6351 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴0))))
5619, 50cnf 20960 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶ 𝐽)
574, 56syl 17 . . . . . . . . 9 (𝜑𝐺:(0[,]1)⟶ 𝐽)
5857feqmptd 6206 . . . . . . . 8 (𝜑𝐺 = (𝑠 ∈ (0[,]1) ↦ (𝐺𝑠)))
5943, 55, 583eqtr4d 2665 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺)
60 cvmliftphtlem.0 . . . . . . 7 (𝜑 → (0𝐴0) = 𝑃)
6137cnmptid 21374 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 𝑠) ∈ (II Cn II))
6223a1i 11 . . . . . . . . . 10 (𝜑 → 0 ∈ (0[,]1))
6337, 37, 62cnmptc 21375 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
6437, 61, 63, 17cnmpt12f 21379 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) ∈ (II Cn 𝐶))
651cvmlift 30986 . . . . . . . . 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 1324 . . . . . . . 8 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
67 coeq2 5240 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝐹𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))))
6867eqeq1d 2623 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝐹𝑓) = 𝐺 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺))
69 fveq1 6147 . . . . . . . . . . . 12 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0))
70 oveq1 6611 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠𝐴0) = (0𝐴0))
71 eqid 2621 . . . . . . . . . . . . . 14 (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))
72 ovex 6632 . . . . . . . . . . . . . 14 (0𝐴0) ∈ V
7370, 71, 72fvmpt 6239 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0))
7423, 73ax-mp 5 . . . . . . . . . . . 12 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))‘0) = (0𝐴0)
7569, 74syl6eq 2671 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (𝑓‘0) = (0𝐴0))
7675eqeq1d 2623 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴0) = 𝑃))
7768, 76anbi12d 746 . . . . . . . . 9 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) → (((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0))) = 𝐺 ∧ (0𝐴0) = 𝑃)))
7877riota2 6587 . . . . . . . 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 692 . . . . . . 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 955 . . . . . 6 (𝜑 → (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
812, 80syl5eq 2667 . . . . 5 (𝜑𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)))
8219, 1cnf 20960 . . . . . . 7 (𝑀 ∈ (II Cn 𝐶) → 𝑀:(0[,]1)⟶𝐵)
838, 82syl 17 . . . . . 6 (𝜑𝑀:(0[,]1)⟶𝐵)
8483feqmptd 6206 . . . . 5 (𝜑𝑀 = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)))
8581, 84eqtr3d 2657 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)))
86 mpteqb 6255 . . . . 5 (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠)))
87 ovex 6632 . . . . . 6 (𝑠𝐴0) ∈ V
8887a1i 11 . . . . 5 (𝑠 ∈ (0[,]1) → (𝑠𝐴0) ∈ V)
8986, 88mprg 2921 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴0)) = (𝑠 ∈ (0[,]1) ↦ (𝑀𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠))
9085, 89sylib 208 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠))
9190r19.21bi 2927 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴0) = (𝑀𝑠))
92 1elunit 12233 . . . . . . . . . . . . . 14 1 ∈ (0[,]1)
93 opelxpi 5108 . . . . . . . . . . . . . 14 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
9492, 93mpan2 706 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1)))
95 fvco3 6232 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨𝑠, 1⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
9622, 94, 95syl2an 494 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐹‘(𝐴‘⟨𝑠, 1⟩)))
9729fveq1d 6150 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨𝑠, 1⟩) = (𝐾‘⟨𝑠, 1⟩))
9896, 97eqtr3d 2657 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨𝑠, 1⟩)) = (𝐾‘⟨𝑠, 1⟩))
99 df-ov 6607 . . . . . . . . . . . 12 (𝑠𝐴1) = (𝐴‘⟨𝑠, 1⟩)
10099fveq2i 6151 . . . . . . . . . . 11 (𝐹‘(𝑠𝐴1)) = (𝐹‘(𝐴‘⟨𝑠, 1⟩))
101 df-ov 6607 . . . . . . . . . . 11 (𝑠𝐾1) = (𝐾‘⟨𝑠, 1⟩)
10298, 100, 1013eqtr4g 2680 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝑠𝐾1))
10340simprd 479 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐾1) = (𝐻𝑠))
104102, 103eqtrd 2655 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑠𝐴1)) = (𝐻𝑠))
105104mpteq2dva 4704 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐻𝑠)))
106 fovrn 6757 . . . . . . . . . . 11 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
10792, 106mp3an3 1410 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
10822, 107sylan 488 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴1) ∈ 𝐵)
109 eqidd 2622 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
110 fveq2 6148 . . . . . . . . 9 (𝑥 = (𝑠𝐴1) → (𝐹𝑥) = (𝐹‘(𝑠𝐴1)))
111108, 109, 53, 110fmptco 6351 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑠𝐴1))))
11219, 50cnf 20960 . . . . . . . . . 10 (𝐻 ∈ (II Cn 𝐽) → 𝐻:(0[,]1)⟶ 𝐽)
11310, 112syl 17 . . . . . . . . 9 (𝜑𝐻:(0[,]1)⟶ 𝐽)
114113feqmptd 6206 . . . . . . . 8 (𝜑𝐻 = (𝑠 ∈ (0[,]1) ↦ (𝐻𝑠)))
115105, 111, 1143eqtr4d 2665 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻)
116 iiconn 22598 . . . . . . . . . . . . 13 II ∈ Conn
117116a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ Conn)
118 iinllyconn 30941 . . . . . . . . . . . . 13 II ∈ 𝑛-Locally Conn
119118a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ 𝑛-Locally Conn)
12037, 63, 61, 17cnmpt12f 21379 . . . . . . . . . . . 12 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) ∈ (II Cn 𝐶))
121 cvmtop1 30947 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
1223, 121syl 17 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ Top)
1231toptopon 20648 . . . . . . . . . . . . . 14 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
124122, 123sylib 208 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (TopOn‘𝐵))
125 ffvelrn 6313 . . . . . . . . . . . . . 14 ((𝑀:(0[,]1)⟶𝐵 ∧ 0 ∈ (0[,]1)) → (𝑀‘0) ∈ 𝐵)
12683, 23, 125sylancl 693 . . . . . . . . . . . . 13 (𝜑 → (𝑀‘0) ∈ 𝐵)
127 cnconst2 20997 . . . . . . . . . . . . 13 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘0) ∈ 𝐵) → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
12837, 124, 126, 127syl3anc 1323 . . . . . . . . . . . 12 (𝜑 → ((0[,]1) × {(𝑀‘0)}) ∈ (II Cn 𝐶))
1294, 10, 11phtpyi 22691 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐾𝑠) = (𝐺‘0) ∧ (1𝐾𝑠) = (𝐺‘1)))
130129simpld 475 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐾𝑠) = (𝐺‘0))
131 opelxpi 5108 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
13223, 131mpan 705 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (0[,]1) → ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
133 fvco3 6232 . . . . . . . . . . . . . . . . . . 19 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨0, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
13422, 132, 133syl2an 494 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐹‘(𝐴‘⟨0, 𝑠⟩)))
13529fveq1d 6150 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨0, 𝑠⟩) = (𝐾‘⟨0, 𝑠⟩))
136134, 135eqtr3d 2657 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨0, 𝑠⟩)) = (𝐾‘⟨0, 𝑠⟩))
137 df-ov 6607 . . . . . . . . . . . . . . . . . 18 (0𝐴𝑠) = (𝐴‘⟨0, 𝑠⟩)
138137fveq2i 6151 . . . . . . . . . . . . . . . . 17 (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝐴‘⟨0, 𝑠⟩))
139 df-ov 6607 . . . . . . . . . . . . . . . . 17 (0𝐾𝑠) = (𝐾‘⟨0, 𝑠⟩)
140136, 138, 1393eqtr4g 2680 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (0𝐾𝑠))
1417simp3d 1073 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀‘0) = 𝑃)
142141adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (0[,]1)) → (𝑀‘0) = 𝑃)
143142fveq2d 6152 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐹𝑃))
1446adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑃) = (𝐺‘0))
145143, 144eqtrd 2655 . . . . . . . . . . . . . . . 16 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝑀‘0)) = (𝐺‘0))
146130, 140, 1453eqtr4d 2665 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(0𝐴𝑠)) = (𝐹‘(𝑀‘0)))
147146mpteq2dva 4704 . . . . . . . . . . . . . 14 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0))))
148 fconstmpt 5123 . . . . . . . . . . . . . 14 ((0[,]1) × {(𝐹‘(𝑀‘0))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘0)))
149147, 148syl6eqr 2673 . . . . . . . . . . . . 13 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
150 fovrn 6757 . . . . . . . . . . . . . . . 16 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
15123, 150mp3an2 1409 . . . . . . . . . . . . . . 15 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
15222, 151sylan 488 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐴𝑠) ∈ 𝐵)
153 eqidd 2622 . . . . . . . . . . . . . 14 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)))
154 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = (0𝐴𝑠) → (𝐹𝑥) = (𝐹‘(0𝐴𝑠)))
155152, 153, 53, 154fmptco 6351 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(0𝐴𝑠))))
156 ffn 6002 . . . . . . . . . . . . . . 15 (𝐹:𝐵 𝐽𝐹 Fn 𝐵)
15752, 156syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐵)
158 fcoconst 6355 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐵 ∧ (𝑀‘0) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
159157, 126, 158syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})) = ((0[,]1) × {(𝐹‘(𝑀‘0))}))
160149, 155, 1593eqtr4d 2665 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘0)})))
16160, 141eqtr4d 2658 . . . . . . . . . . . . 13 (𝜑 → (0𝐴0) = (𝑀‘0))
162 oveq2 6612 . . . . . . . . . . . . . . 15 (𝑠 = 0 → (0𝐴𝑠) = (0𝐴0))
163 eqid 2621 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))
164162, 163, 72fvmpt 6239 . . . . . . . . . . . . . 14 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0))
16523, 164ax-mp 5 . . . . . . . . . . . . 13 ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (0𝐴0)
166 fvex 6158 . . . . . . . . . . . . . . 15 (𝑀‘0) ∈ V
167166fvconst2 6423 . . . . . . . . . . . . . 14 (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0))
16823, 167ax-mp 5 . . . . . . . . . . . . 13 (((0[,]1) × {(𝑀‘0)})‘0) = (𝑀‘0)
169161, 165, 1683eqtr4g 2680 . . . . . . . . . . . 12 (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘0)})‘0))
1701, 19, 3, 117, 119, 62, 120, 128, 160, 169cvmliftmoi 30970 . . . . . . . . . . 11 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = ((0[,]1) × {(𝑀‘0)}))
171 fconstmpt 5123 . . . . . . . . . . 11 ((0[,]1) × {(𝑀‘0)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0))
172170, 171syl6eq 2671 . . . . . . . . . 10 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)))
173 mpteqb 6255 . . . . . . . . . . 11 (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0)))
174 ovex 6632 . . . . . . . . . . . 12 (0𝐴𝑠) ∈ V
175174a1i 11 . . . . . . . . . . 11 (𝑠 ∈ (0[,]1) → (0𝐴𝑠) ∈ V)
176173, 175mprg 2921 . . . . . . . . . 10 ((𝑠 ∈ (0[,]1) ↦ (0𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘0)) ↔ ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
177172, 176sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0))
178 oveq2 6612 . . . . . . . . . . 11 (𝑠 = 1 → (0𝐴𝑠) = (0𝐴1))
179178eqeq1d 2623 . . . . . . . . . 10 (𝑠 = 1 → ((0𝐴𝑠) = (𝑀‘0) ↔ (0𝐴1) = (𝑀‘0)))
180179rspcv 3291 . . . . . . . . 9 (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(0𝐴𝑠) = (𝑀‘0) → (0𝐴1) = (𝑀‘0)))
18192, 177, 180mpsyl 68 . . . . . . . 8 (𝜑 → (0𝐴1) = (𝑀‘0))
182181, 141eqtrd 2655 . . . . . . 7 (𝜑 → (0𝐴1) = 𝑃)
18392a1i 11 . . . . . . . . . 10 (𝜑 → 1 ∈ (0[,]1))
18437, 37, 183cnmptc 21375 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (0[,]1) ↦ 1) ∈ (II Cn II))
18537, 61, 184, 17cnmpt12f 21379 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) ∈ (II Cn 𝐶))
1861cvmlift 30986 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽)) ∧ (𝑃𝐵 ∧ (𝐹𝑃) = (𝐻‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
1873, 10, 5, 14, 186syl22anc 1324 . . . . . . . 8 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
188 coeq2 5240 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝐹𝑓) = (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
189188eqeq1d 2623 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝐹𝑓) = 𝐻 ↔ (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻))
190 fveq1 6147 . . . . . . . . . . . 12 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0))
191 oveq1 6611 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠𝐴1) = (0𝐴1))
192 eqid 2621 . . . . . . . . . . . . . 14 (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))
193 ovex 6632 . . . . . . . . . . . . . 14 (0𝐴1) ∈ V
194191, 192, 193fvmpt 6239 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1))
19523, 194ax-mp 5 . . . . . . . . . . . 12 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))‘0) = (0𝐴1)
196190, 195syl6eq 2671 . . . . . . . . . . 11 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (𝑓‘0) = (0𝐴1))
197196eqeq1d 2623 . . . . . . . . . 10 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → ((𝑓‘0) = 𝑃 ↔ (0𝐴1) = 𝑃))
198189, 197anbi12d 746 . . . . . . . . 9 (𝑓 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) → (((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃)))
199198riota2 6587 . . . . . . . 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))))
200185, 187, 199syl2anc 692 . . . . . . 7 (𝜑 → (((𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))) = 𝐻 ∧ (0𝐴1) = 𝑃) ↔ (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1))))
201115, 182, 200mpbi2and 955 . . . . . 6 (𝜑 → (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
2029, 201syl5eq 2667 . . . . 5 (𝜑𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)))
20319, 1cnf 20960 . . . . . . 7 (𝑁 ∈ (II Cn 𝐶) → 𝑁:(0[,]1)⟶𝐵)
20416, 203syl 17 . . . . . 6 (𝜑𝑁:(0[,]1)⟶𝐵)
205204feqmptd 6206 . . . . 5 (𝜑𝑁 = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)))
206202, 205eqtr3d 2657 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)))
207 mpteqb 6255 . . . . 5 (∀𝑠 ∈ (0[,]1)(𝑠𝐴1) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠)))
208 ovex 6632 . . . . . 6 (𝑠𝐴1) ∈ V
209208a1i 11 . . . . 5 (𝑠 ∈ (0[,]1) → (𝑠𝐴1) ∈ V)
210207, 209mprg 2921 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (𝑠𝐴1)) = (𝑠 ∈ (0[,]1) ↦ (𝑁𝑠)) ↔ ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠))
211206, 210sylib 208 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(𝑠𝐴1) = (𝑁𝑠))
212211r19.21bi 2927 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐴1) = (𝑁𝑠))
213177r19.21bi 2927 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐴𝑠) = (𝑀‘0))
21437, 184, 61, 17cnmpt12f 21379 . . . . . 6 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) ∈ (II Cn 𝐶))
215 ffvelrn 6313 . . . . . . . 8 ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → (𝑀‘1) ∈ 𝐵)
21683, 92, 215sylancl 693 . . . . . . 7 (𝜑 → (𝑀‘1) ∈ 𝐵)
217 cnconst2 20997 . . . . . . 7 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ (𝑀‘1) ∈ 𝐵) → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
21837, 124, 216, 217syl3anc 1323 . . . . . 6 (𝜑 → ((0[,]1) × {(𝑀‘1)}) ∈ (II Cn 𝐶))
219 opelxpi 5108 . . . . . . . . . . . . . 14 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
22092, 219mpan 705 . . . . . . . . . . . . 13 (𝑠 ∈ (0[,]1) → ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1)))
221 fvco3 6232 . . . . . . . . . . . . 13 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ ⟨1, 𝑠⟩ ∈ ((0[,]1) × (0[,]1))) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
22222, 220, 221syl2an 494 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐹‘(𝐴‘⟨1, 𝑠⟩)))
22329fveq1d 6150 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝐴)‘⟨1, 𝑠⟩) = (𝐾‘⟨1, 𝑠⟩))
224222, 223eqtr3d 2657 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(𝐴‘⟨1, 𝑠⟩)) = (𝐾‘⟨1, 𝑠⟩))
225 df-ov 6607 . . . . . . . . . . . 12 (1𝐴𝑠) = (𝐴‘⟨1, 𝑠⟩)
226225fveq2i 6151 . . . . . . . . . . 11 (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝐴‘⟨1, 𝑠⟩))
227 df-ov 6607 . . . . . . . . . . 11 (1𝐾𝑠) = (𝐾‘⟨1, 𝑠⟩)
228224, 226, 2273eqtr4g 2680 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (1𝐾𝑠))
229129simprd 479 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐾𝑠) = (𝐺‘1))
2307simp2d 1072 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑀) = 𝐺)
231230adantr 481 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑀) = 𝐺)
232231fveq1d 6150 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐺‘1))
23383adantr 481 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0[,]1)) → 𝑀:(0[,]1)⟶𝐵)
234 fvco3 6232 . . . . . . . . . . . 12 ((𝑀:(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐹‘(𝑀‘1)))
235233, 92, 234sylancl 693 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (0[,]1)) → ((𝐹𝑀)‘1) = (𝐹‘(𝑀‘1)))
236232, 235eqtr3d 2657 . . . . . . . . . 10 ((𝜑𝑠 ∈ (0[,]1)) → (𝐺‘1) = (𝐹‘(𝑀‘1)))
237228, 229, 2363eqtrd 2659 . . . . . . . . 9 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘(1𝐴𝑠)) = (𝐹‘(𝑀‘1)))
238237mpteq2dva 4704 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1))))
239 fconstmpt 5123 . . . . . . . 8 ((0[,]1) × {(𝐹‘(𝑀‘1))}) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(𝑀‘1)))
240238, 239syl6eqr 2673 . . . . . . 7 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
241 fovrn 6757 . . . . . . . . . 10 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵 ∧ 1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
24292, 241mp3an2 1409 . . . . . . . . 9 ((𝐴:((0[,]1) × (0[,]1))⟶𝐵𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
24322, 242sylan 488 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐴𝑠) ∈ 𝐵)
244 eqidd 2622 . . . . . . . 8 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)))
245 fveq2 6148 . . . . . . . 8 (𝑥 = (1𝐴𝑠) → (𝐹𝑥) = (𝐹‘(1𝐴𝑠)))
246243, 244, 53, 245fmptco 6351 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝑠 ∈ (0[,]1) ↦ (𝐹‘(1𝐴𝑠))))
247 fcoconst 6355 . . . . . . . 8 ((𝐹 Fn 𝐵 ∧ (𝑀‘1) ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
248157, 216, 247syl2anc 692 . . . . . . 7 (𝜑 → (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})) = ((0[,]1) × {(𝐹‘(𝑀‘1))}))
249240, 246, 2483eqtr4d 2665 . . . . . 6 (𝜑 → (𝐹 ∘ (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))) = (𝐹 ∘ ((0[,]1) × {(𝑀‘1)})))
250 oveq1 6611 . . . . . . . . . 10 (𝑠 = 1 → (𝑠𝐴0) = (1𝐴0))
251 fveq2 6148 . . . . . . . . . 10 (𝑠 = 1 → (𝑀𝑠) = (𝑀‘1))
252250, 251eqeq12d 2636 . . . . . . . . 9 (𝑠 = 1 → ((𝑠𝐴0) = (𝑀𝑠) ↔ (1𝐴0) = (𝑀‘1)))
253252rspcv 3291 . . . . . . . 8 (1 ∈ (0[,]1) → (∀𝑠 ∈ (0[,]1)(𝑠𝐴0) = (𝑀𝑠) → (1𝐴0) = (𝑀‘1)))
25492, 90, 253mpsyl 68 . . . . . . 7 (𝜑 → (1𝐴0) = (𝑀‘1))
255 oveq2 6612 . . . . . . . . 9 (𝑠 = 0 → (1𝐴𝑠) = (1𝐴0))
256 eqid 2621 . . . . . . . . 9 (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))
257 ovex 6632 . . . . . . . . 9 (1𝐴0) ∈ V
258255, 256, 257fvmpt 6239 . . . . . . . 8 (0 ∈ (0[,]1) → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0))
25923, 258ax-mp 5 . . . . . . 7 ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (1𝐴0)
260 fvex 6158 . . . . . . . . 9 (𝑀‘1) ∈ V
261260fvconst2 6423 . . . . . . . 8 (0 ∈ (0[,]1) → (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1))
26223, 261ax-mp 5 . . . . . . 7 (((0[,]1) × {(𝑀‘1)})‘0) = (𝑀‘1)
263254, 259, 2623eqtr4g 2680 . . . . . 6 (𝜑 → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠))‘0) = (((0[,]1) × {(𝑀‘1)})‘0))
2641, 19, 3, 117, 119, 62, 214, 218, 249, 263cvmliftmoi 30970 . . . . 5 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = ((0[,]1) × {(𝑀‘1)}))
265 fconstmpt 5123 . . . . 5 ((0[,]1) × {(𝑀‘1)}) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1))
266264, 265syl6eq 2671 . . . 4 (𝜑 → (𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)))
267 mpteqb 6255 . . . . 5 (∀𝑠 ∈ (0[,]1)(1𝐴𝑠) ∈ V → ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1)))
268 ovex 6632 . . . . . 6 (1𝐴𝑠) ∈ V
269268a1i 11 . . . . 5 (𝑠 ∈ (0[,]1) → (1𝐴𝑠) ∈ V)
270267, 269mprg 2921 . . . 4 ((𝑠 ∈ (0[,]1) ↦ (1𝐴𝑠)) = (𝑠 ∈ (0[,]1) ↦ (𝑀‘1)) ↔ ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
271266, 270sylib 208 . . 3 (𝜑 → ∀𝑠 ∈ (0[,]1)(1𝐴𝑠) = (𝑀‘1))
272271r19.21bi 2927 . 2 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐴𝑠) = (𝑀‘1))
2738, 16, 17, 91, 212, 213, 272isphtpy2d 22694 1 (𝜑𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  ∃!wreu 2909  Vcvv 3186  {csn 4148  cop 4154   cuni 4402  cmpt 4673   × cxp 5072  ccom 5078   Fn wfn 5842  wf 5843  cfv 5847  crio 6564  (class class class)co 6604  0cc0 9880  1c1 9881  [,]cicc 12120  Topctop 20617  TopOnctopon 20618   Cn ccn 20938  Conncconn 21124  𝑛-Locally cnlly 21178   ×t ctx 21273  IIcii 22586   Htpy chtpy 22674  PHtpycphtpy 22675   CovMap ccvm 30942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-cn 20941  df-cnp 20942  df-cmp 21100  df-conn 21125  df-lly 21179  df-nlly 21180  df-tx 21275  df-hmeo 21468  df-xms 22035  df-ms 22036  df-tms 22037  df-ii 22588  df-htpy 22677  df-phtpy 22678  df-phtpc 22699  df-pconn 30908  df-sconn 30909  df-cvm 30943
This theorem is referenced by:  cvmliftpht  31005
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