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Theorem cvmlift2lem9a 31411
Description: Lemma for cvmlift2 31424 and cvmlift3 31436. (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 31368 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐶 ∈ Top)
4 cnrest2r 21139 . . 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)
7 ffn 6083 . . . . . 6 (𝐻:𝑌𝐵𝐻 Fn 𝑌)
86, 7syl 17 . . . . 5 (𝜑𝐻 Fn 𝑌)
9 cvmlift2lem9a.4 . . . . 5 (𝜑𝑀𝑌)
10 fnssres 6042 . . . . 5 ((𝐻 Fn 𝑌𝑀𝑌) → (𝐻𝑀) Fn 𝑀)
118, 9, 10syl2anc 694 . . . 4 (𝜑 → (𝐻𝑀) Fn 𝑀)
12 df-ima 5156 . . . . 5 (𝐻𝑀) = ran (𝐻𝑀)
13 cvmlift2lem9a.6 . . . . 5 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1412, 13syl5eqssr 3683 . . . 4 (𝜑 → ran (𝐻𝑀) ⊆ 𝑊)
15 df-f 5930 . . . 4 ((𝐻𝑀):𝑀𝑊 ↔ ((𝐻𝑀) Fn 𝑀 ∧ ran (𝐻𝑀) ⊆ 𝑊))
1611, 14, 15sylanbrc 699 . . 3 (𝜑 → (𝐻𝑀):𝑀𝑊)
17 cvmlift2lem9a.2 . . . . . . . . . . 11 (𝜑𝑇 ∈ (𝑆𝐴))
18 cvmlift2lem9a.3 . . . . . . . . . . . 12 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
1918simpld 474 . . . . . . . . . . 11 (𝜑𝑊𝑇)
20 cvmlift2lem9a.s . . . . . . . . . . . 12 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
2120cvmsf1o 31380 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → (𝐹𝑊):𝑊1-1-onto𝐴)
221, 17, 19, 21syl3anc 1366 . . . . . . . . . 10 (𝜑 → (𝐹𝑊):𝑊1-1-onto𝐴)
2322adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑊):𝑊1-1-onto𝐴)
24 f1of1 6174 . . . . . . . . 9 ((𝐹𝑊):𝑊1-1-onto𝐴 → (𝐹𝑊):𝑊1-1𝐴)
2523, 24syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑊):𝑊1-1𝐴)
26 cvmlift2lem9a.b . . . . . . . . . . . 12 𝐵 = 𝐶
2726toptopon 20770 . . . . . . . . . . 11 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
283, 27sylib 208 . . . . . . . . . 10 (𝜑𝐶 ∈ (TopOn‘𝐵))
2920cvmsss 31375 . . . . . . . . . . . . 13 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
3017, 29syl 17 . . . . . . . . . . . 12 (𝜑𝑇𝐶)
3130, 19sseldd 3637 . . . . . . . . . . 11 (𝜑𝑊𝐶)
32 toponss 20779 . . . . . . . . . . 11 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐶) → 𝑊𝐵)
3328, 31, 32syl2anc 694 . . . . . . . . . 10 (𝜑𝑊𝐵)
34 resttopon 21013 . . . . . . . . . 10 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐵) → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
3528, 33, 34syl2anc 694 . . . . . . . . 9 (𝜑 → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
36 toponss 20779 . . . . . . . . 9 (((𝐶t 𝑊) ∈ (TopOn‘𝑊) ∧ 𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
3735, 36sylan 487 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
38 f1imacnv 6191 . . . . . . . 8 (((𝐹𝑊):𝑊1-1𝐴𝑥𝑊) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
3925, 37, 38syl2anc 694 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
4039imaeq2d 5501 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = ((𝐻𝑀) “ 𝑥))
41 imaco 5678 . . . . . . 7 (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)))
42 cnvco 5340 . . . . . . . . 9 ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐻𝑀) ∘ (𝐹𝑊))
43 cores 5676 . . . . . . . . . . . . 13 (ran (𝐻𝑀) ⊆ 𝑊 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
4414, 43syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
45 resco 5677 . . . . . . . . . . . 12 ((𝐹𝐻) ↾ 𝑀) = (𝐹 ∘ (𝐻𝑀))
4644, 45syl6eqr 2703 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4746adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4847cnveqd 5330 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4942, 48syl5eqr 2699 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) ∘ (𝐹𝑊)) = ((𝐹𝐻) ↾ 𝑀))
5049imaeq1d 5500 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5141, 50syl5eqr 2699 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5240, 51eqtr3d 2687 . . . . 5 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ 𝑥) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
53 cvmlift2lem9a.g . . . . . . . 8 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
54 cvmlift2lem9a.y . . . . . . . . 9 𝑌 = 𝐾
5554cnrest 21137 . . . . . . . 8 (((𝐹𝐻) ∈ (𝐾 Cn 𝐽) ∧ 𝑀𝑌) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5653, 9, 55syl2anc 694 . . . . . . 7 (𝜑 → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5756adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
58 resima2 5467 . . . . . . . 8 (𝑥𝑊 → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
5937, 58syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
601adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
61 restopn2 21029 . . . . . . . . . 10 ((𝐶 ∈ Top ∧ 𝑊𝐶) → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
623, 31, 61syl2anc 694 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
6362simprbda 652 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝐶)
64 cvmopn 31388 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝐶) → (𝐹𝑥) ∈ 𝐽)
6560, 63, 64syl2anc 694 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑥) ∈ 𝐽)
6659, 65eqeltrd 2730 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) ∈ 𝐽)
67 cnima 21117 . . . . . 6 ((((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽) ∧ ((𝐹𝑊) “ 𝑥) ∈ 𝐽) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6857, 66, 67syl2anc 694 . . . . 5 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6952, 68eqeltrd 2730 . . . 4 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
7069ralrimiva 2995 . . 3 (𝜑 → ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
71 cvmlift2lem9a.k . . . . . 6 (𝜑𝐾 ∈ Top)
7254toptopon 20770 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
7371, 72sylib 208 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
74 resttopon 21013 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
7573, 9, 74syl2anc 694 . . . 4 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
76 iscn 21087 . . . 4 (((𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ (𝐶t 𝑊) ∈ (TopOn‘𝑊)) → ((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ↔ ((𝐻𝑀):𝑀𝑊 ∧ ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))))
7775, 35, 76syl2anc 694 . . 3 (𝜑 → ((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ↔ ((𝐻𝑀):𝑀𝑊 ∧ ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))))
7816, 70, 77mpbir2and 977 . 2 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)))
795, 78sseldd 3637 1 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762  ccnv 5142  ran crn 5144  cres 5145  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  Homeochmeo 21604   CovMap ccvm 31363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-hmeo 21606  df-cvm 31364
This theorem is referenced by:  cvmlift2lem9  31419  cvmlift3lem7  31433
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