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Theorem cvmlift2lem9a 35288
Description: Lemma for cvmlift2 35301 and cvmlift3 35313. (Contributed by Mario Carneiro, 9-Jul-2015.)
Hypotheses
Ref Expression
cvmlift2lem9a.b 𝐵 = 𝐶
cvmlift2lem9a.y 𝑌 = 𝐾
cvmlift2lem9a.s 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
cvmlift2lem9a.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift2lem9a.h (𝜑𝐻:𝑌𝐵)
cvmlift2lem9a.g (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
cvmlift2lem9a.k (𝜑𝐾 ∈ Top)
cvmlift2lem9a.1 (𝜑𝑋𝑌)
cvmlift2lem9a.2 (𝜑𝑇 ∈ (𝑆𝐴))
cvmlift2lem9a.3 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
cvmlift2lem9a.4 (𝜑𝑀𝑌)
cvmlift2lem9a.6 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
Assertion
Ref Expression
cvmlift2lem9a (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
Distinct variable groups:   𝑐,𝑑,𝑘,𝑠,𝐴   𝐹,𝑐,𝑑,𝑘,𝑠   𝐽,𝑐,𝑑,𝑘,𝑠   𝑇,𝑐,𝑑,𝑠   𝐶,𝑐,𝑑,𝑘,𝑠   𝑊,𝑐,𝑑
Allowed substitution hints:   𝜑(𝑘,𝑠,𝑐,𝑑)   𝐵(𝑘,𝑠,𝑐,𝑑)   𝑆(𝑘,𝑠,𝑐,𝑑)   𝑇(𝑘)   𝐻(𝑘,𝑠,𝑐,𝑑)   𝐾(𝑘,𝑠,𝑐,𝑑)   𝑀(𝑘,𝑠,𝑐,𝑑)   𝑊(𝑘,𝑠)   𝑋(𝑘,𝑠,𝑐,𝑑)   𝑌(𝑘,𝑠,𝑐,𝑑)

Proof of Theorem cvmlift2lem9a
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cvmlift2lem9a.f . . . 4 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
2 cvmtop1 35245 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐶 ∈ Top)
4 cnrest2r 23311 . . 3 (𝐶 ∈ Top → ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ⊆ ((𝐾t 𝑀) Cn 𝐶))
53, 4syl 17 . 2 (𝜑 → ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ⊆ ((𝐾t 𝑀) Cn 𝐶))
6 cvmlift2lem9a.h . . . . . 6 (𝜑𝐻:𝑌𝐵)
76ffnd 6738 . . . . 5 (𝜑𝐻 Fn 𝑌)
8 cvmlift2lem9a.4 . . . . 5 (𝜑𝑀𝑌)
9 fnssres 6692 . . . . 5 ((𝐻 Fn 𝑌𝑀𝑌) → (𝐻𝑀) Fn 𝑀)
107, 8, 9syl2anc 584 . . . 4 (𝜑 → (𝐻𝑀) Fn 𝑀)
11 df-ima 5702 . . . . 5 (𝐻𝑀) = ran (𝐻𝑀)
12 cvmlift2lem9a.6 . . . . 5 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1311, 12eqsstrrid 4045 . . . 4 (𝜑 → ran (𝐻𝑀) ⊆ 𝑊)
14 df-f 6567 . . . 4 ((𝐻𝑀):𝑀𝑊 ↔ ((𝐻𝑀) Fn 𝑀 ∧ ran (𝐻𝑀) ⊆ 𝑊))
1510, 13, 14sylanbrc 583 . . 3 (𝜑 → (𝐻𝑀):𝑀𝑊)
16 cvmlift2lem9a.2 . . . . . . . . . . 11 (𝜑𝑇 ∈ (𝑆𝐴))
17 cvmlift2lem9a.3 . . . . . . . . . . . 12 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
1817simpld 494 . . . . . . . . . . 11 (𝜑𝑊𝑇)
19 cvmlift2lem9a.s . . . . . . . . . . . 12 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
2019cvmsf1o 35257 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → (𝐹𝑊):𝑊1-1-onto𝐴)
211, 16, 18, 20syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝐹𝑊):𝑊1-1-onto𝐴)
2221adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑊):𝑊1-1-onto𝐴)
23 f1of1 6848 . . . . . . . . 9 ((𝐹𝑊):𝑊1-1-onto𝐴 → (𝐹𝑊):𝑊1-1𝐴)
2422, 23syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑊):𝑊1-1𝐴)
25 cvmlift2lem9a.b . . . . . . . . . . . 12 𝐵 = 𝐶
2625toptopon 22939 . . . . . . . . . . 11 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
273, 26sylib 218 . . . . . . . . . 10 (𝜑𝐶 ∈ (TopOn‘𝐵))
2819cvmsss 35252 . . . . . . . . . . . . 13 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
2916, 28syl 17 . . . . . . . . . . . 12 (𝜑𝑇𝐶)
3029, 18sseldd 3996 . . . . . . . . . . 11 (𝜑𝑊𝐶)
31 toponss 22949 . . . . . . . . . . 11 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐶) → 𝑊𝐵)
3227, 30, 31syl2anc 584 . . . . . . . . . 10 (𝜑𝑊𝐵)
33 resttopon 23185 . . . . . . . . . 10 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐵) → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
3427, 32, 33syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
35 toponss 22949 . . . . . . . . 9 (((𝐶t 𝑊) ∈ (TopOn‘𝑊) ∧ 𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
3634, 35sylan 580 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
37 f1imacnv 6865 . . . . . . . 8 (((𝐹𝑊):𝑊1-1𝐴𝑥𝑊) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
3824, 36, 37syl2anc 584 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
3938imaeq2d 6080 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = ((𝐻𝑀) “ 𝑥))
40 imaco 6273 . . . . . . 7 (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)))
41 cnvco 5899 . . . . . . . . 9 ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐻𝑀) ∘ (𝐹𝑊))
42 cores 6271 . . . . . . . . . . . . 13 (ran (𝐻𝑀) ⊆ 𝑊 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
4313, 42syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
44 resco 6272 . . . . . . . . . . . 12 ((𝐹𝐻) ↾ 𝑀) = (𝐹 ∘ (𝐻𝑀))
4543, 44eqtr4di 2793 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4645adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4746cnveqd 5889 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4841, 47eqtr3id 2789 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) ∘ (𝐹𝑊)) = ((𝐹𝐻) ↾ 𝑀))
4948imaeq1d 6079 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5040, 49eqtr3id 2789 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5139, 50eqtr3d 2777 . . . . 5 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ 𝑥) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
52 cvmlift2lem9a.g . . . . . . . 8 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
53 cvmlift2lem9a.y . . . . . . . . 9 𝑌 = 𝐾
5453cnrest 23309 . . . . . . . 8 (((𝐹𝐻) ∈ (𝐾 Cn 𝐽) ∧ 𝑀𝑌) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5552, 8, 54syl2anc 584 . . . . . . 7 (𝜑 → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5655adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
57 resima2 6036 . . . . . . . 8 (𝑥𝑊 → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
5836, 57syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
591adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
60 restopn2 23201 . . . . . . . . . 10 ((𝐶 ∈ Top ∧ 𝑊𝐶) → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
613, 30, 60syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
6261simprbda 498 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝐶)
63 cvmopn 35265 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝐶) → (𝐹𝑥) ∈ 𝐽)
6459, 62, 63syl2anc 584 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑥) ∈ 𝐽)
6558, 64eqeltrd 2839 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) ∈ 𝐽)
66 cnima 23289 . . . . . 6 ((((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽) ∧ ((𝐹𝑊) “ 𝑥) ∈ 𝐽) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6756, 65, 66syl2anc 584 . . . . 5 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6851, 67eqeltrd 2839 . . . 4 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
6968ralrimiva 3144 . . 3 (𝜑 → ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
70 cvmlift2lem9a.k . . . . . 6 (𝜑𝐾 ∈ Top)
7153toptopon 22939 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
7270, 71sylib 218 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
73 resttopon 23185 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
7472, 8, 73syl2anc 584 . . . 4 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
75 iscn 23259 . . . 4 (((𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ (𝐶t 𝑊) ∈ (TopOn‘𝑊)) → ((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ↔ ((𝐻𝑀):𝑀𝑊 ∧ ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))))
7674, 34, 75syl2anc 584 . . 3 (𝜑 → ((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ↔ ((𝐻𝑀):𝑀𝑊 ∧ ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))))
7715, 69, 76mpbir2and 713 . 2 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)))
785, 77sseldd 3996 1 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  ran crn 5690  cres 5691  cima 5692  ccom 5693   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Homeochmeo 23777   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
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-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-op 4638  df-uni 4913  df-int 4952  df-iun 4998  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-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-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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-hmeo 23779  df-cvm 35241
This theorem is referenced by:  cvmlift2lem9  35296  cvmlift3lem7  35310
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