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Theorem cvmlift3lem7 30561
 Description: Lemma for cvmlift3 30564. (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 ∈ SCon)
cvmlift3.l (𝜑𝐾 ∈ 𝑛-Locally PCon)
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) ∈ PCon)
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 ∈ SCon)
6 cvmlift3.l . . . . 5 (𝜑𝐾 ∈ 𝑛-Locally PCon)
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 30557 . . . 4 (𝜑𝐻:𝑌𝐵)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 30559 . . . . 5 (𝜑 → (𝐹𝐻) = 𝐺)
1413, 8eqeltrd 2688 . . . 4 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
15 scontop 30464 . . . . 5 (𝐾 ∈ SCon → 𝐾 ∈ Top)
165, 15syl 17 . . . 4 (𝜑𝐾 ∈ Top)
17 cvmlift3lem7.3 . . . . . 6 (𝜑𝑀 ⊆ (𝐺𝐴))
18 cnvimass 5404 . . . . . . 7 (𝐺𝐴) ⊆ dom 𝐺
19 eqid 2610 . . . . . . . . 9 𝐽 = 𝐽
202, 19cnf 20860 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
21 fdm 5964 . . . . . . . 8 (𝐺:𝑌 𝐽 → dom 𝐺 = 𝑌)
228, 20, 213syl 18 . . . . . . 7 (𝜑 → dom 𝐺 = 𝑌)
2318, 22syl5sseq 3616 . . . . . 6 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
2417, 23sstrd 3578 . . . . 5 (𝜑𝑀𝑌)
25 cvmlift3lem7.5 . . . . . 6 (𝜑𝑉𝑀)
26 cvmlift3lem7.6 . . . . . 6 (𝜑𝑋𝑉)
2725, 26sseldd 3569 . . . . 5 (𝜑𝑋𝑀)
2824, 27sseldd 3569 . . . 4 (𝜑𝑋𝑌)
29 cvmlift3lem7.2 . . . 4 (𝜑𝑇 ∈ (𝑆𝐴))
3012, 28ffvelrnd 6268 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝐵)
31 fvco3 6185 . . . . . . . 8 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3212, 28, 31syl2anc 691 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3313fveq1d 6105 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
3432, 33eqtr3d 2646 . . . . . 6 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
35 cvmlift3lem7.1 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
3634, 35eqeltrd 2688 . . . . 5 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
37 cvmlift3lem7.w . . . . . 6 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
383, 1, 37cvmsiota 30513 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
394, 29, 30, 36, 38syl13anc 1320 . . . 4 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
40 eqid 2610 . . . . . . . . . . 11 (𝐻𝑋) = (𝐻𝑋)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 30558 . . . . . . . . . . 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 222 . . . . . . . . . 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 699 . . . . . . . . 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 6102 . . . . . . . . . . 11 (𝑓 = → (𝑓‘0) = (‘0))
4645eqeq1d 2612 . . . . . . . . . 10 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
47 fveq1 6102 . . . . . . . . . . 11 (𝑓 = → (𝑓‘1) = (‘1))
4847eqeq1d 2612 . . . . . . . . . 10 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
49 coeq2 5202 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝐺𝑓) = (𝐺))
5049eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
5150anbi1d 737 . . . . . . . . . . . . . 14 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5251riotabidv 6513 . . . . . . . . . . . . 13 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
53 coeq2 5202 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝐹𝑎) = (𝐹𝑔))
5453eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺) ↔ (𝐹𝑔) = (𝐺)))
55 fveq1 6102 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
5655eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
5754, 56anbi12d 743 . . . . . . . . . . . . . 14 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5857cbvriotav 6522 . . . . . . . . . . . . 13 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))
5952, 58syl6eqr 2662 . . . . . . . . . . . 12 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)))
6059fveq1d 6105 . . . . . . . . . . 11 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1))
6160eqeq1d 2612 . . . . . . . . . 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 1391 . . . . . . . . 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))))
6362cbvrexv 3148 . . . . . . . 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 207 . . . . . . 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) ∈ PCon)
6665adantr 480 . . . . . . . 8 ((𝜑𝑦𝑀) → (𝐾t 𝑀) ∈ PCon)
672restuni 20776 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀𝑌) → 𝑀 = (𝐾t 𝑀))
6816, 24, 67syl2anc 691 . . . . . . . . . 10 (𝜑𝑀 = (𝐾t 𝑀))
6927, 68eleqtrd 2690 . . . . . . . . 9 (𝜑𝑋 (𝐾t 𝑀))
7069adantr 480 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑋 (𝐾t 𝑀))
7168eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑦𝑀𝑦 (𝐾t 𝑀)))
7271biimpa 500 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑦 (𝐾t 𝑀))
73 eqid 2610 . . . . . . . . 9 (𝐾t 𝑀) = (𝐾t 𝑀)
7473pconcn 30460 . . . . . . . 8 (((𝐾t 𝑀) ∈ PCon ∧ 𝑋 (𝐾t 𝑀) ∧ 𝑦 (𝐾t 𝑀)) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
7566, 70, 72, 74syl3anc 1318 . . . . . . 7 ((𝜑𝑦𝑀) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76 reeanv 3086 . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 ∈ SCon)
796ad3antrrr 762 . . . . . . . . . . 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 PCon)
807ad3antrrr 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 762 . . . . . . . . . . 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 795 . . . . . . . . . . 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 796 . . . . . . . . . . 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 790 . . . . . . . . . . 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 797 . . . . . . . . . . 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 792 . . . . . . . . . . 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 2612 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺𝑛) ↔ (𝐹𝑔) = (𝐺𝑛)))
9455eqeq1d 2612 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻𝑋) ↔ (𝑔‘0) = (𝐻𝑋)))
9593, 94anbi12d 743 . . . . . . . . . . . 12 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺𝑛) ∧ (𝑎‘0) = (𝐻𝑋)) ↔ ((𝐹𝑔) = (𝐺𝑛) ∧ (𝑔‘0) = (𝐻𝑋))))
9695cbvriotav 6522 . . . . . . . . . . 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 30560 . . . . . . . . . 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 449 . . . . . . . . 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 3020 . . . . . . . 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, 99syl5bir 232 . . . . . . 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 711 . . . . . 6 ((𝜑𝑦𝑀) → (𝐻𝑦) ∈ 𝑊)
102101ralrimiva 2949 . . . . 5 (𝜑 → ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊)
103 ffun 5961 . . . . . . 7 (𝐻:𝑌𝐵 → Fun 𝐻)
10412, 103syl 17 . . . . . 6 (𝜑 → Fun 𝐻)
105 fdm 5964 . . . . . . . 8 (𝐻:𝑌𝐵 → dom 𝐻 = 𝑌)
10612, 105syl 17 . . . . . . 7 (𝜑 → dom 𝐻 = 𝑌)
10724, 106sseqtr4d 3605 . . . . . 6 (𝜑𝑀 ⊆ dom 𝐻)
108 funimass4 6157 . . . . . 6 ((Fun 𝐻𝑀 ⊆ dom 𝐻) → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
109104, 107, 108syl2anc 691 . . . . 5 (𝜑 → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
110102, 109mpbird 246 . . . 4 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1111, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 110cvmlift2lem9a 30539 . . 3 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
11273cncnpi 20892 . . 3 (((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶) ∧ 𝑋 (𝐾t 𝑀)) → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
113111, 69, 112syl2anc 691 . 2 (𝜑 → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
114 cvmlift3lem7.4 . . . . 5 (𝜑𝑉𝐾)
1152ssntr 20672 . . . . 5 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑉𝐾𝑉𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
11616, 24, 114, 25, 115syl22anc 1319 . . . 4 (𝜑𝑉 ⊆ ((int‘𝐾)‘𝑀))
117116, 26sseldd 3569 . . 3 (𝜑𝑋 ∈ ((int‘𝐾)‘𝑀))
1182, 1cnprest 20903 . . 3 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑋 ∈ ((int‘𝐾)‘𝑀) ∧ 𝐻:𝑌𝐵)) → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
11916, 24, 117, 12, 118syl22anc 1319 . 2 (𝜑 → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
120113, 119mpbird 246 1 (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  0cc0 9815  1c1 9816   ↾t crest 15904  Topctop 20517  intcnt 20631   Cn ccn 20838   CnP ccnp 20839  𝑛-Locally cnlly 21078  Homeochmeo 21366  IIcii 22486  PConcpcon 30455  SConcscon 30456   CovMap ccvm 30491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cn 20841  df-cnp 20842  df-cmp 21000  df-con 21025  df-lly 21079  df-nlly 21080  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-ii 22488  df-htpy 22577  df-phtpy 22578  df-phtpc 22599  df-pco 22613  df-pcon 30457  df-scon 30458  df-cvm 30492 This theorem is referenced by:  cvmlift3lem8  30562
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