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Theorem cvmlift3lem7 31015
Description: Lemma for cvmlift3 31018. (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 31011 . . . 4 (𝜑𝐻:𝑌𝐵)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 31013 . . . . 5 (𝜑 → (𝐹𝐻) = 𝐺)
1413, 8eqeltrd 2698 . . . 4 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 30918 . . . . 5 (𝐾 ∈ SConn → 𝐾 ∈ Top)
165, 15syl 17 . . . 4 (𝜑𝐾 ∈ Top)
17 cvmlift3lem7.3 . . . . . 6 (𝜑𝑀 ⊆ (𝐺𝐴))
18 cnvimass 5444 . . . . . . 7 (𝐺𝐴) ⊆ dom 𝐺
19 eqid 2621 . . . . . . . . 9 𝐽 = 𝐽
202, 19cnf 20960 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
21 fdm 6008 . . . . . . . 8 (𝐺:𝑌 𝐽 → dom 𝐺 = 𝑌)
228, 20, 213syl 18 . . . . . . 7 (𝜑 → dom 𝐺 = 𝑌)
2318, 22syl5sseq 3632 . . . . . 6 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
2417, 23sstrd 3593 . . . . 5 (𝜑𝑀𝑌)
25 cvmlift3lem7.5 . . . . . 6 (𝜑𝑉𝑀)
26 cvmlift3lem7.6 . . . . . 6 (𝜑𝑋𝑉)
2725, 26sseldd 3584 . . . . 5 (𝜑𝑋𝑀)
2824, 27sseldd 3584 . . . 4 (𝜑𝑋𝑌)
29 cvmlift3lem7.2 . . . 4 (𝜑𝑇 ∈ (𝑆𝐴))
3012, 28ffvelrnd 6316 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝐵)
31 fvco3 6232 . . . . . . . 8 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3212, 28, 31syl2anc 692 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3313fveq1d 6150 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
3432, 33eqtr3d 2657 . . . . . 6 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
35 cvmlift3lem7.1 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
3634, 35eqeltrd 2698 . . . . 5 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
37 cvmlift3lem7.w . . . . . 6 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
383, 1, 37cvmsiota 30967 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
394, 29, 30, 36, 38syl13anc 1325 . . . 4 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
40 eqid 2621 . . . . . . . . . . 11 (𝐻𝑋) = (𝐻𝑋)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 31012 . . . . . . . . . . 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 223 . . . . . . . . . 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 701 . . . . . . . . 9 (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
4443adantr 481 . . . . . . . 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 6147 . . . . . . . . . . 11 (𝑓 = → (𝑓‘0) = (‘0))
4645eqeq1d 2623 . . . . . . . . . 10 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
47 fveq1 6147 . . . . . . . . . . 11 (𝑓 = → (𝑓‘1) = (‘1))
4847eqeq1d 2623 . . . . . . . . . 10 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
49 coeq2 5240 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝐺𝑓) = (𝐺))
5049eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
5150anbi1d 740 . . . . . . . . . . . . . 14 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5251riotabidv 6567 . . . . . . . . . . . . 13 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
53 coeq2 5240 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝐹𝑎) = (𝐹𝑔))
5453eqeq1d 2623 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺) ↔ (𝐹𝑔) = (𝐺)))
55 fveq1 6147 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
5655eqeq1d 2623 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
5754, 56anbi12d 746 . . . . . . . . . . . . . 14 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5857cbvriotav 6576 . . . . . . . . . . . . 13 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))
5952, 58syl6eqr 2673 . . . . . . . . . . . 12 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)))
6059fveq1d 6150 . . . . . . . . . . 11 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1))
6160eqeq1d 2623 . . . . . . . . . 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 1396 . . . . . . . . 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 3160 . . . . . . . 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 208 . . . . . . 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 481 . . . . . . . 8 ((𝜑𝑦𝑀) → (𝐾t 𝑀) ∈ PConn)
672restuni 20876 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀𝑌) → 𝑀 = (𝐾t 𝑀))
6816, 24, 67syl2anc 692 . . . . . . . . . 10 (𝜑𝑀 = (𝐾t 𝑀))
6927, 68eleqtrd 2700 . . . . . . . . 9 (𝜑𝑋 (𝐾t 𝑀))
7069adantr 481 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑋 (𝐾t 𝑀))
7168eleq2d 2684 . . . . . . . . 9 (𝜑 → (𝑦𝑀𝑦 (𝐾t 𝑀)))
7271biimpa 501 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑦 (𝐾t 𝑀))
73 eqid 2621 . . . . . . . . 9 (𝐾t 𝑀) = (𝐾t 𝑀)
7473pconncn 30914 . . . . . . . 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 1323 . . . . . . 7 ((𝜑𝑦𝑀) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76 reeanv 3097 . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 765 . . . . . . . . . . 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 798 . . . . . . . . . . 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 799 . . . . . . . . . . 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 793 . . . . . . . . . . 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 800 . . . . . . . . . . 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 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))) → ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
9353eqeq1d 2623 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺𝑛) ↔ (𝐹𝑔) = (𝐺𝑛)))
9455eqeq1d 2623 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻𝑋) ↔ (𝑔‘0) = (𝐻𝑋)))
9593, 94anbi12d 746 . . . . . . . . . . . 12 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺𝑛) ∧ (𝑎‘0) = (𝐻𝑋)) ↔ ((𝐹𝑔) = (𝐺𝑛) ∧ (𝑔‘0) = (𝐻𝑋))))
9695cbvriotav 6576 . . . . . . . . . . 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 31014 . . . . . . . . . 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 450 . . . . . . . . 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 3031 . . . . . . . 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 233 . . . . . . 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 714 . . . . . 6 ((𝜑𝑦𝑀) → (𝐻𝑦) ∈ 𝑊)
102101ralrimiva 2960 . . . . 5 (𝜑 → ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊)
103 ffun 6005 . . . . . . 7 (𝐻:𝑌𝐵 → Fun 𝐻)
10412, 103syl 17 . . . . . 6 (𝜑 → Fun 𝐻)
105 fdm 6008 . . . . . . . 8 (𝐻:𝑌𝐵 → dom 𝐻 = 𝑌)
10612, 105syl 17 . . . . . . 7 (𝜑 → dom 𝐻 = 𝑌)
10724, 106sseqtr4d 3621 . . . . . 6 (𝜑𝑀 ⊆ dom 𝐻)
108 funimass4 6204 . . . . . 6 ((Fun 𝐻𝑀 ⊆ dom 𝐻) → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
109104, 107, 108syl2anc 692 . . . . 5 (𝜑 → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
110102, 109mpbird 247 . . . 4 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1111, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 110cvmlift2lem9a 30993 . . 3 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
11273cncnpi 20992 . . 3 (((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶) ∧ 𝑋 (𝐾t 𝑀)) → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
113111, 69, 112syl2anc 692 . 2 (𝜑 → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
114 cvmlift3lem7.4 . . . . 5 (𝜑𝑉𝐾)
1152ssntr 20772 . . . . 5 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑉𝐾𝑉𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
11616, 24, 114, 25, 115syl22anc 1324 . . . 4 (𝜑𝑉 ⊆ ((int‘𝐾)‘𝑀))
117116, 26sseldd 3584 . . 3 (𝜑𝑋 ∈ ((int‘𝐾)‘𝑀))
1182, 1cnprest 21003 . . 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 1324 . 2 (𝜑 → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
120113, 119mpbird 247 1 (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402  cmpt 4673  ccnv 5073  dom cdm 5074  cres 5076  cima 5077  ccom 5078  Fun wfun 5841  wf 5843  cfv 5847  crio 6564  (class class class)co 6604  0cc0 9880  1c1 9881  t crest 16002  Topctop 20617  intcnt 20731   Cn ccn 20938   CnP ccnp 20939  𝑛-Locally cnlly 21178  Homeochmeo 21466  IIcii 22586  PConncpconn 30909  SConncsconn 30910   CovMap ccvm 30945
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-pco 22713  df-pconn 30911  df-sconn 30912  df-cvm 30946
This theorem is referenced by:  cvmlift3lem8  31016
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