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Theorem cvmlift3lem7 35310
Description: Lemma for cvmlift3 35313. (Contributed by Mario Carneiro, 9-Jul-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))))})
cvmlift3lem7.1 (𝜑 → (𝐺𝑋) ∈ 𝐴)
cvmlift3lem7.2 (𝜑𝑇 ∈ (𝑆𝐴))
cvmlift3lem7.3 (𝜑𝑀 ⊆ (𝐺𝐴))
cvmlift3lem7.w 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
cvmlift3lem7.7 (𝜑 → (𝐾t 𝑀) ∈ PConn)
cvmlift3lem7.4 (𝜑𝑉𝐾)
cvmlift3lem7.5 (𝜑𝑉𝑀)
cvmlift3lem7.6 (𝜑𝑋𝑉)
Assertion
Ref Expression
cvmlift3lem7 (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Distinct variable groups:   𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝐴   𝑓,𝑔,𝑧,𝑏,𝑥   𝐽,𝑏   𝑔,𝑐,𝑥,𝐽,𝑑,𝑓,𝑘,𝑠   𝐹,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠   𝑥,𝑧,𝐹   𝑓,𝑀,𝑔,𝑥   𝐻,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑆,𝑏,𝑓,𝑥   𝐵,𝑏,𝑑,𝑓,𝑔,𝑥,𝑧   𝑋,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝐺,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧   𝑇,𝑏,𝑐,𝑑,𝑠   𝐶,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠,𝑥,𝑧   𝜑,𝑓,𝑥   𝐾,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧   𝑃,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑂,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧   𝑊,𝑐,𝑑,𝑓,𝑥
Allowed substitution hints:   𝜑(𝑧,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑)   𝐴(𝑥,𝑔)   𝐵(𝑘,𝑠,𝑐)   𝑃(𝑘,𝑠)   𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝑇(𝑥,𝑧,𝑓,𝑔,𝑘)   𝐺(𝑠)   𝐻(𝑘,𝑠)   𝐽(𝑧)   𝐾(𝑘,𝑠,𝑑)   𝑀(𝑧,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑂(𝑘,𝑠,𝑑)   𝑉(𝑥,𝑧,𝑓,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑊(𝑧,𝑔,𝑘,𝑠,𝑏)   𝑋(𝑘,𝑠)   𝑌(𝑘,𝑠,𝑏,𝑐,𝑑)

Proof of Theorem cvmlift3lem7
Dummy variables 𝑎 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift3.b . . . 4 𝐵 = 𝐶
2 cvmlift3.y . . . 4 𝑌 = 𝐾
3 cvmlift3lem7.s . . . 4 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
4 cvmlift3.f . . . 4 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
5 cvmlift3.k . . . . 5 (𝜑𝐾 ∈ SConn)
6 cvmlift3.l . . . . 5 (𝜑𝐾 ∈ 𝑛-Locally PConn)
7 cvmlift3.o . . . . 5 (𝜑𝑂𝑌)
8 cvmlift3.g . . . . 5 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
9 cvmlift3.p . . . . 5 (𝜑𝑃𝐵)
10 cvmlift3.e . . . . 5 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
11 cvmlift3.h . . . . 5 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
121, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem3 35306 . . . 4 (𝜑𝐻:𝑌𝐵)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 35308 . . . . 5 (𝜑 → (𝐹𝐻) = 𝐺)
1413, 8eqeltrd 2839 . . . 4 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 35213 . . . . 5 (𝐾 ∈ SConn → 𝐾 ∈ Top)
165, 15syl 17 . . . 4 (𝜑𝐾 ∈ Top)
17 cvmlift3lem7.3 . . . . . 6 (𝜑𝑀 ⊆ (𝐺𝐴))
18 cnvimass 6102 . . . . . . 7 (𝐺𝐴) ⊆ dom 𝐺
19 eqid 2735 . . . . . . . . 9 𝐽 = 𝐽
202, 19cnf 23270 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
21 fdm 6746 . . . . . . . 8 (𝐺:𝑌 𝐽 → dom 𝐺 = 𝑌)
228, 20, 213syl 18 . . . . . . 7 (𝜑 → dom 𝐺 = 𝑌)
2318, 22sseqtrid 4048 . . . . . 6 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
2417, 23sstrd 4006 . . . . 5 (𝜑𝑀𝑌)
25 cvmlift3lem7.5 . . . . . 6 (𝜑𝑉𝑀)
26 cvmlift3lem7.6 . . . . . 6 (𝜑𝑋𝑉)
2725, 26sseldd 3996 . . . . 5 (𝜑𝑋𝑀)
2824, 27sseldd 3996 . . . 4 (𝜑𝑋𝑌)
29 cvmlift3lem7.2 . . . 4 (𝜑𝑇 ∈ (𝑆𝐴))
3012, 28ffvelcdmd 7105 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝐵)
31 fvco3 7008 . . . . . . . 8 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3212, 28, 31syl2anc 584 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3313fveq1d 6909 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
3432, 33eqtr3d 2777 . . . . . 6 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
35 cvmlift3lem7.1 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
3634, 35eqeltrd 2839 . . . . 5 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
37 cvmlift3lem7.w . . . . . 6 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
383, 1, 37cvmsiota 35262 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
394, 29, 30, 36, 38syl13anc 1371 . . . 4 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
40 eqid 2735 . . . . . . . . . . 11 (𝐻𝑋) = (𝐻𝑋)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 35307 . . . . . . . . . . 11 ((𝜑𝑋𝑌) → ((𝐻𝑋) = (𝐻𝑋) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋))))
4240, 41mpbii 233 . . . . . . . . . 10 ((𝜑𝑋𝑌) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
4328, 42mpdan 687 . . . . . . . . 9 (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
4443adantr 480 . . . . . . . 8 ((𝜑𝑦𝑀) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
45 fveq1 6906 . . . . . . . . . . 11 (𝑓 = → (𝑓‘0) = (‘0))
4645eqeq1d 2737 . . . . . . . . . 10 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
47 fveq1 6906 . . . . . . . . . . 11 (𝑓 = → (𝑓‘1) = (‘1))
4847eqeq1d 2737 . . . . . . . . . 10 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
49 coeq2 5872 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝐺𝑓) = (𝐺))
5049eqeq2d 2746 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
5150anbi1d 631 . . . . . . . . . . . . . 14 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5251riotabidv 7390 . . . . . . . . . . . . 13 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
53 coeq2 5872 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝐹𝑎) = (𝐹𝑔))
5453eqeq1d 2737 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺) ↔ (𝐹𝑔) = (𝐺)))
55 fveq1 6906 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
5655eqeq1d 2737 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
5754, 56anbi12d 632 . . . . . . . . . . . . . 14 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5857cbvriotavw 7398 . . . . . . . . . . . . 13 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))
5952, 58eqtr4di 2793 . . . . . . . . . . . 12 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)))
6059fveq1d 6909 . . . . . . . . . . 11 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1))
6160eqeq1d 2737 . . . . . . . . . 10 (𝑓 = → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋) ↔ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
6246, 48, 613anbi123d 1435 . . . . . . . . 9 (𝑓 = → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)) ↔ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋))))
6362cbvrexvw 3236 . . . . . . . 8 (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)) ↔ ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
6444, 63sylib 218 . . . . . . 7 ((𝜑𝑦𝑀) → ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
65 cvmlift3lem7.7 . . . . . . . . 9 (𝜑 → (𝐾t 𝑀) ∈ PConn)
6665adantr 480 . . . . . . . 8 ((𝜑𝑦𝑀) → (𝐾t 𝑀) ∈ PConn)
672restuni 23186 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀𝑌) → 𝑀 = (𝐾t 𝑀))
6816, 24, 67syl2anc 584 . . . . . . . . . 10 (𝜑𝑀 = (𝐾t 𝑀))
6927, 68eleqtrd 2841 . . . . . . . . 9 (𝜑𝑋 (𝐾t 𝑀))
7069adantr 480 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑋 (𝐾t 𝑀))
7168eleq2d 2825 . . . . . . . . 9 (𝜑 → (𝑦𝑀𝑦 (𝐾t 𝑀)))
7271biimpa 476 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑦 (𝐾t 𝑀))
73 eqid 2735 . . . . . . . . 9 (𝐾t 𝑀) = (𝐾t 𝑀)
7473pconncn 35209 . . . . . . . 8 (((𝐾t 𝑀) ∈ PConn ∧ 𝑋 (𝐾t 𝑀) ∧ 𝑦 (𝐾t 𝑀)) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
7566, 70, 72, 74syl3anc 1370 . . . . . . 7 ((𝜑𝑦𝑀) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76 reeanv 3227 . . . . . . . 8 (∃ ∈ (II Cn 𝐾)∃𝑛 ∈ (II Cn (𝐾t 𝑀))(((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) ↔ (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)))
774ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
785ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐾 ∈ SConn)
796ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐾 ∈ 𝑛-Locally PConn)
807ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑂𝑌)
818ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐺 ∈ (𝐾 Cn 𝐽))
829ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑃𝐵)
8310ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐹𝑃) = (𝐺𝑂))
8435ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐺𝑋) ∈ 𝐴)
8529ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑇 ∈ (𝑆𝐴))
8617ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑀 ⊆ (𝐺𝐴))
8727ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑋𝑀)
88 simpllr 776 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑦𝑀)
89 simplrl 777 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ∈ (II Cn 𝐾))
90 simprl 771 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
91 simplrr 778 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑛 ∈ (II Cn (𝐾t 𝑀)))
92 simprr 773 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
9353eqeq1d 2737 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺𝑛) ↔ (𝐹𝑔) = (𝐺𝑛)))
9455eqeq1d 2737 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻𝑋) ↔ (𝑔‘0) = (𝐻𝑋)))
9593, 94anbi12d 632 . . . . . . . . . . . 12 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺𝑛) ∧ (𝑎‘0) = (𝐻𝑋)) ↔ ((𝐹𝑔) = (𝐺𝑛) ∧ (𝑔‘0) = (𝐻𝑋))))
9695cbvriotavw 7398 . . . . . . . . . . 11 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺𝑛) ∧ (𝑎‘0) = (𝐻𝑋))) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑛) ∧ (𝑔‘0) = (𝐻𝑋)))
971, 2, 77, 78, 79, 80, 81, 82, 83, 11, 3, 84, 85, 86, 37, 87, 88, 89, 58, 90, 91, 92, 96cvmlift3lem6 35309 . . . . . . . . . 10 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐻𝑦) ∈ 𝑊)
9897ex 412 . . . . . . . . 9 (((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) → ((((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻𝑦) ∈ 𝑊))
9998rexlimdvva 3211 . . . . . . . 8 ((𝜑𝑦𝑀) → (∃ ∈ (II Cn 𝐾)∃𝑛 ∈ (II Cn (𝐾t 𝑀))(((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻𝑦) ∈ 𝑊))
10076, 99biimtrrid 243 . . . . . . 7 ((𝜑𝑦𝑀) → ((∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻𝑦) ∈ 𝑊))
10164, 75, 100mp2and 699 . . . . . 6 ((𝜑𝑦𝑀) → (𝐻𝑦) ∈ 𝑊)
102101ralrimiva 3144 . . . . 5 (𝜑 → ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊)
10312ffund 6741 . . . . . 6 (𝜑 → Fun 𝐻)
10412fdmd 6747 . . . . . . 7 (𝜑 → dom 𝐻 = 𝑌)
10524, 104sseqtrrd 4037 . . . . . 6 (𝜑𝑀 ⊆ dom 𝐻)
106 funimass4 6973 . . . . . 6 ((Fun 𝐻𝑀 ⊆ dom 𝐻) → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
107103, 105, 106syl2anc 584 . . . . 5 (𝜑 → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
108102, 107mpbird 257 . . . 4 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1091, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 108cvmlift2lem9a 35288 . . 3 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
11073cncnpi 23302 . . 3 (((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶) ∧ 𝑋 (𝐾t 𝑀)) → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
111109, 69, 110syl2anc 584 . 2 (𝜑 → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
112 cvmlift3lem7.4 . . . . 5 (𝜑𝑉𝐾)
1132ssntr 23082 . . . . 5 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑉𝐾𝑉𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
11416, 24, 112, 25, 113syl22anc 839 . . . 4 (𝜑𝑉 ⊆ ((int‘𝐾)‘𝑀))
115114, 26sseldd 3996 . . 3 (𝜑𝑋 ∈ ((int‘𝐾)‘𝑀))
1162, 1cnprest 23313 . . 3 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑋 ∈ ((int‘𝐾)‘𝑀) ∧ 𝐻:𝑌𝐵)) → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
11716, 24, 115, 12, 116syl22anc 839 . 2 (𝜑 → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
118111, 117mpbird 257 1 (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cdif 3960  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  dom cdm 5689  cres 5691  cima 5692  ccom 5693  Fun wfun 6557  wf 6559  cfv 6563  crio 7387  (class class class)co 7431  0cc0 11153  1c1 11154  t crest 17467  Topctop 22915  intcnt 23041   Cn ccn 23248   CnP ccnp 23249  𝑛-Locally cnlly 23489  Homeochmeo 23777  IIcii 24915  PConncpconn 35204  SConncsconn 35205   CovMap ccvm 35240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cn 23251  df-cnp 23252  df-cmp 23411  df-conn 23436  df-lly 23490  df-nlly 23491  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348  df-ii 24917  df-cncf 24918  df-htpy 25016  df-phtpy 25017  df-phtpc 25038  df-pco 25052  df-pconn 35206  df-sconn 35207  df-cvm 35241
This theorem is referenced by:  cvmlift3lem8  35311
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