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Theorem cvmlift3lem9 31294
 Description: Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
cvmlift3.b 𝐵 = 𝐶
cvmlift3.y 𝑌 = 𝐾
cvmlift3.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k (𝜑𝐾 ∈ SConn)
cvmlift3.l (𝜑𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o (𝜑𝑂𝑌)
cvmlift3.g (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p (𝜑𝑃𝐵)
cvmlift3.e (𝜑 → (𝐹𝑃) = (𝐺𝑂))
cvmlift3.h 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
cvmlift3lem7.s 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmlift3lem9 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Distinct variable groups:   𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝑔,𝑥   𝐽,𝑐   𝑔,𝑑,𝑥,𝐽,𝑓,𝑘,𝑠   𝐹,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠   𝑥,𝑧,𝐹   𝐻,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑆,𝑓,𝑥   𝐵,𝑑,𝑓,𝑔,𝑥,𝑧   𝐺,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧   𝐶,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠,𝑥,𝑧   𝜑,𝑓,𝑥   𝐾,𝑐,𝑓,𝑔,𝑥,𝑧   𝑃,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑂,𝑐,𝑓,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝐵(𝑘,𝑠,𝑐)   𝑃(𝑘,𝑠)   𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝐺(𝑠)   𝐻(𝑘,𝑠)   𝐽(𝑧)   𝐾(𝑘,𝑠,𝑑)   𝑂(𝑘,𝑠,𝑑)   𝑌(𝑘,𝑠,𝑐,𝑑)

Proof of Theorem cvmlift3lem9
StepHypRef Expression
1 cvmlift3.b . . 3 𝐵 = 𝐶
2 cvmlift3.y . . 3 𝑌 = 𝐾
3 cvmlift3.f . . 3 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmlift3.k . . 3 (𝜑𝐾 ∈ SConn)
5 cvmlift3.l . . 3 (𝜑𝐾 ∈ 𝑛-Locally PConn)
6 cvmlift3.o . . 3 (𝜑𝑂𝑌)
7 cvmlift3.g . . 3 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
8 cvmlift3.p . . 3 (𝜑𝑃𝐵)
9 cvmlift3.e . . 3 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
10 cvmlift3.h . . 3 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
11 cvmlift3lem7.s . . 3 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem8 31293 . 2 (𝜑𝐻 ∈ (𝐾 Cn 𝐶))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 31290 . 2 (𝜑 → (𝐹𝐻) = 𝐺)
14 iitopon 22676 . . . . . 6 II ∈ (TopOn‘(0[,]1))
1514a1i 11 . . . . 5 (𝜑 → II ∈ (TopOn‘(0[,]1)))
16 sconntop 31195 . . . . . . 7 (𝐾 ∈ SConn → 𝐾 ∈ Top)
174, 16syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
182toptopon 20716 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
1917, 18sylib 208 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
20 cnconst2 21081 . . . . 5 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑂𝑌) → ((0[,]1) × {𝑂}) ∈ (II Cn 𝐾))
2115, 19, 6, 20syl3anc 1325 . . . 4 (𝜑 → ((0[,]1) × {𝑂}) ∈ (II Cn 𝐾))
22 0elunit 12287 . . . . 5 0 ∈ (0[,]1)
23 fvconst2g 6464 . . . . 5 ((𝑂𝑌 ∧ 0 ∈ (0[,]1)) → (((0[,]1) × {𝑂})‘0) = 𝑂)
246, 22, 23sylancl 694 . . . 4 (𝜑 → (((0[,]1) × {𝑂})‘0) = 𝑂)
25 1elunit 12288 . . . . 5 1 ∈ (0[,]1)
26 fvconst2g 6464 . . . . 5 ((𝑂𝑌 ∧ 1 ∈ (0[,]1)) → (((0[,]1) × {𝑂})‘1) = 𝑂)
276, 25, 26sylancl 694 . . . 4 (𝜑 → (((0[,]1) × {𝑂})‘1) = 𝑂)
289sneqd 4187 . . . . . . . . 9 (𝜑 → {(𝐹𝑃)} = {(𝐺𝑂)})
2928xpeq2d 5137 . . . . . . . 8 (𝜑 → ((0[,]1) × {(𝐹𝑃)}) = ((0[,]1) × {(𝐺𝑂)}))
30 cvmcn 31229 . . . . . . . . . 10 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
31 eqid 2621 . . . . . . . . . . 11 𝐽 = 𝐽
321, 31cnf 21044 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
33 ffn 6043 . . . . . . . . . 10 (𝐹:𝐵 𝐽𝐹 Fn 𝐵)
343, 30, 32, 334syl 19 . . . . . . . . 9 (𝜑𝐹 Fn 𝐵)
35 fcoconst 6398 . . . . . . . . 9 ((𝐹 Fn 𝐵𝑃𝐵) → (𝐹 ∘ ((0[,]1) × {𝑃})) = ((0[,]1) × {(𝐹𝑃)}))
3634, 8, 35syl2anc 693 . . . . . . . 8 (𝜑 → (𝐹 ∘ ((0[,]1) × {𝑃})) = ((0[,]1) × {(𝐹𝑃)}))
372, 31cnf 21044 . . . . . . . . . . 11 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
387, 37syl 17 . . . . . . . . . 10 (𝜑𝐺:𝑌 𝐽)
39 ffn 6043 . . . . . . . . . 10 (𝐺:𝑌 𝐽𝐺 Fn 𝑌)
4038, 39syl 17 . . . . . . . . 9 (𝜑𝐺 Fn 𝑌)
41 fcoconst 6398 . . . . . . . . 9 ((𝐺 Fn 𝑌𝑂𝑌) → (𝐺 ∘ ((0[,]1) × {𝑂})) = ((0[,]1) × {(𝐺𝑂)}))
4240, 6, 41syl2anc 693 . . . . . . . 8 (𝜑 → (𝐺 ∘ ((0[,]1) × {𝑂})) = ((0[,]1) × {(𝐺𝑂)}))
4329, 36, 423eqtr4d 2665 . . . . . . 7 (𝜑 → (𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})))
44 fvconst2g 6464 . . . . . . . 8 ((𝑃𝐵 ∧ 0 ∈ (0[,]1)) → (((0[,]1) × {𝑃})‘0) = 𝑃)
458, 22, 44sylancl 694 . . . . . . 7 (𝜑 → (((0[,]1) × {𝑃})‘0) = 𝑃)
46 cvmtop1 31227 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
473, 46syl 17 . . . . . . . . . 10 (𝜑𝐶 ∈ Top)
481toptopon 20716 . . . . . . . . . 10 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
4947, 48sylib 208 . . . . . . . . 9 (𝜑𝐶 ∈ (TopOn‘𝐵))
50 cnconst2 21081 . . . . . . . . 9 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ 𝑃𝐵) → ((0[,]1) × {𝑃}) ∈ (II Cn 𝐶))
5115, 49, 8, 50syl3anc 1325 . . . . . . . 8 (𝜑 → ((0[,]1) × {𝑃}) ∈ (II Cn 𝐶))
52 cvmtop2 31228 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
533, 52syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
5431toptopon 20716 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
5553, 54sylib 208 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
5638, 6ffvelrnd 6358 . . . . . . . . . . 11 (𝜑 → (𝐺𝑂) ∈ 𝐽)
57 cnconst2 21081 . . . . . . . . . . 11 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐺𝑂) ∈ 𝐽) → ((0[,]1) × {(𝐺𝑂)}) ∈ (II Cn 𝐽))
5815, 55, 56, 57syl3anc 1325 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐺𝑂)}) ∈ (II Cn 𝐽))
5942, 58eqeltrd 2700 . . . . . . . . 9 (𝜑 → (𝐺 ∘ ((0[,]1) × {𝑂})) ∈ (II Cn 𝐽))
60 fvconst2g 6464 . . . . . . . . . . 11 (((𝐺𝑂) ∈ 𝐽 ∧ 0 ∈ (0[,]1)) → (((0[,]1) × {(𝐺𝑂)})‘0) = (𝐺𝑂))
6156, 22, 60sylancl 694 . . . . . . . . . 10 (𝜑 → (((0[,]1) × {(𝐺𝑂)})‘0) = (𝐺𝑂))
6242fveq1d 6191 . . . . . . . . . 10 (𝜑 → ((𝐺 ∘ ((0[,]1) × {𝑂}))‘0) = (((0[,]1) × {(𝐺𝑂)})‘0))
6361, 62, 93eqtr4rd 2666 . . . . . . . . 9 (𝜑 → (𝐹𝑃) = ((𝐺 ∘ ((0[,]1) × {𝑂}))‘0))
641cvmlift 31266 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺 ∘ ((0[,]1) × {𝑂})) ∈ (II Cn 𝐽)) ∧ (𝑃𝐵 ∧ (𝐹𝑃) = ((𝐺 ∘ ((0[,]1) × {𝑂}))‘0))) → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))
653, 59, 8, 63, 64syl22anc 1326 . . . . . . . 8 (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))
66 coeq2 5278 . . . . . . . . . . 11 (𝑔 = ((0[,]1) × {𝑃}) → (𝐹𝑔) = (𝐹 ∘ ((0[,]1) × {𝑃})))
6766eqeq1d 2623 . . . . . . . . . 10 (𝑔 = ((0[,]1) × {𝑃}) → ((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ↔ (𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂}))))
68 fveq1 6188 . . . . . . . . . . 11 (𝑔 = ((0[,]1) × {𝑃}) → (𝑔‘0) = (((0[,]1) × {𝑃})‘0))
6968eqeq1d 2623 . . . . . . . . . 10 (𝑔 = ((0[,]1) × {𝑃}) → ((𝑔‘0) = 𝑃 ↔ (((0[,]1) × {𝑃})‘0) = 𝑃))
7067, 69anbi12d 747 . . . . . . . . 9 (𝑔 = ((0[,]1) × {𝑃}) → (((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (((0[,]1) × {𝑃})‘0) = 𝑃)))
7170riota2 6630 . . . . . . . 8 ((((0[,]1) × {𝑃}) ∈ (II Cn 𝐶) ∧ ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) → (((𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (((0[,]1) × {𝑃})‘0) = 𝑃) ↔ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) = ((0[,]1) × {𝑃})))
7251, 65, 71syl2anc 693 . . . . . . 7 (𝜑 → (((𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (((0[,]1) × {𝑃})‘0) = 𝑃) ↔ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) = ((0[,]1) × {𝑃})))
7343, 45, 72mpbi2and 956 . . . . . 6 (𝜑 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) = ((0[,]1) × {𝑃}))
7473fveq1d 6191 . . . . 5 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = (((0[,]1) × {𝑃})‘1))
75 fvconst2g 6464 . . . . . 6 ((𝑃𝐵 ∧ 1 ∈ (0[,]1)) → (((0[,]1) × {𝑃})‘1) = 𝑃)
768, 25, 75sylancl 694 . . . . 5 (𝜑 → (((0[,]1) × {𝑃})‘1) = 𝑃)
7774, 76eqtrd 2655 . . . 4 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)
78 fveq1 6188 . . . . . . 7 (𝑓 = ((0[,]1) × {𝑂}) → (𝑓‘0) = (((0[,]1) × {𝑂})‘0))
7978eqeq1d 2623 . . . . . 6 (𝑓 = ((0[,]1) × {𝑂}) → ((𝑓‘0) = 𝑂 ↔ (((0[,]1) × {𝑂})‘0) = 𝑂))
80 fveq1 6188 . . . . . . 7 (𝑓 = ((0[,]1) × {𝑂}) → (𝑓‘1) = (((0[,]1) × {𝑂})‘1))
8180eqeq1d 2623 . . . . . 6 (𝑓 = ((0[,]1) × {𝑂}) → ((𝑓‘1) = 𝑂 ↔ (((0[,]1) × {𝑂})‘1) = 𝑂))
82 coeq2 5278 . . . . . . . . . . 11 (𝑓 = ((0[,]1) × {𝑂}) → (𝐺𝑓) = (𝐺 ∘ ((0[,]1) × {𝑂})))
8382eqeq2d 2631 . . . . . . . . . 10 (𝑓 = ((0[,]1) × {𝑂}) → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂}))))
8483anbi1d 741 . . . . . . . . 9 (𝑓 = ((0[,]1) × {𝑂}) → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)))
8584riotabidv 6610 . . . . . . . 8 (𝑓 = ((0[,]1) × {𝑂}) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)))
8685fveq1d 6191 . . . . . . 7 (𝑓 = ((0[,]1) × {𝑂}) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1))
8786eqeq1d 2623 . . . . . 6 (𝑓 = ((0[,]1) × {𝑂}) → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃))
8879, 81, 873anbi123d 1398 . . . . 5 (𝑓 = ((0[,]1) × {𝑂}) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃) ↔ ((((0[,]1) × {𝑂})‘0) = 𝑂 ∧ (((0[,]1) × {𝑂})‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)))
8988rspcev 3307 . . . 4 ((((0[,]1) × {𝑂}) ∈ (II Cn 𝐾) ∧ ((((0[,]1) × {𝑂})‘0) = 𝑂 ∧ (((0[,]1) × {𝑂})‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃))
9021, 24, 27, 77, 89syl13anc 1327 . . 3 (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃))
911, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem4 31289 . . . 4 ((𝜑𝑂𝑌) → ((𝐻𝑂) = 𝑃 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)))
926, 91mpdan 702 . . 3 (𝜑 → ((𝐻𝑂) = 𝑃 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)))
9390, 92mpbird 247 . 2 (𝜑 → (𝐻𝑂) = 𝑃)
94 coeq2 5278 . . . . 5 (𝑓 = 𝐻 → (𝐹𝑓) = (𝐹𝐻))
9594eqeq1d 2623 . . . 4 (𝑓 = 𝐻 → ((𝐹𝑓) = 𝐺 ↔ (𝐹𝐻) = 𝐺))
96 fveq1 6188 . . . . 5 (𝑓 = 𝐻 → (𝑓𝑂) = (𝐻𝑂))
9796eqeq1d 2623 . . . 4 (𝑓 = 𝐻 → ((𝑓𝑂) = 𝑃 ↔ (𝐻𝑂) = 𝑃))
9895, 97anbi12d 747 . . 3 (𝑓 = 𝐻 → (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ((𝐹𝐻) = 𝐺 ∧ (𝐻𝑂) = 𝑃)))
9998rspcev 3307 . 2 ((𝐻 ∈ (𝐾 Cn 𝐶) ∧ ((𝐹𝐻) = 𝐺 ∧ (𝐻𝑂) = 𝑃)) → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
10012, 13, 93, 99syl12anc 1323 1 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989  ∀wral 2911  ∃wrex 2912  ∃!wreu 2913  {crab 2915   ∖ cdif 3569   ∩ cin 3571  ∅c0 3913  𝒫 cpw 4156  {csn 4175  ∪ cuni 4434   ↦ cmpt 4727   × cxp 5110  ◡ccnv 5111   ↾ cres 5114   “ cima 5115   ∘ ccom 5116   Fn wfn 5881  ⟶wf 5882  ‘cfv 5886  ℩crio 6607  (class class class)co 6647  0cc0 9933  1c1 9934  [,]cicc 12175   ↾t crest 16075  Topctop 20692  TopOnctopon 20709   Cn ccn 21022  𝑛-Locally cnlly 21262  Homeochmeo 21550  IIcii 22672  PConncpconn 31186  SConncsconn 31187   CovMap ccvm 31222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011  ax-addf 10012  ax-mulf 10013 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894  df-om 7063  df-1st 7165  df-2nd 7166  df-supp 7293  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-ec 7741  df-map 7856  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fsupp 8273  df-fi 8314  df-sup 8345  df-inf 8346  df-oi 8412  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-z 11375  df-dec 11491  df-uz 11685  df-q 11786  df-rp 11830  df-xneg 11943  df-xadd 11944  df-xmul 11945  df-ioo 12176  df-ico 12178  df-icc 12179  df-fz 12324  df-fzo 12462  df-fl 12588  df-seq 12797  df-exp 12856  df-hash 13113  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-clim 14213  df-sum 14411  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-starv 15950  df-sca 15951  df-vsca 15952  df-ip 15953  df-tset 15954  df-ple 15955  df-ds 15958  df-unif 15959  df-hom 15960  df-cco 15961  df-rest 16077  df-topn 16078  df-0g 16096  df-gsum 16097  df-topgen 16098  df-pt 16099  df-prds 16102  df-xrs 16156  df-qtop 16161  df-imas 16162  df-xps 16164  df-mre 16240  df-mrc 16241  df-acs 16243  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-submnd 17330  df-mulg 17535  df-cntz 17744  df-cmn 18189  df-psmet 19732  df-xmet 19733  df-met 19734  df-bl 19735  df-mopn 19736  df-cnfld 19741  df-top 20693  df-topon 20710  df-topsp 20731  df-bases 20744  df-cld 20817  df-ntr 20818  df-cls 20819  df-nei 20896  df-cn 21025  df-cnp 21026  df-cmp 21184  df-conn 21209  df-lly 21263  df-nlly 21264  df-tx 21359  df-hmeo 21552  df-xms 22119  df-ms 22120  df-tms 22121  df-ii 22674  df-htpy 22763  df-phtpy 22764  df-phtpc 22785  df-pco 22799  df-pconn 31188  df-sconn 31189  df-cvm 31223 This theorem is referenced by:  cvmlift3  31295
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