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Theorem cvmlift2lem9a 31014
Description: Lemma for cvmlift2 31027 and cvmlift3 31039. (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 30971 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐶 ∈ Top)
4 cnrest2r 21004 . . 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 6004 . . . . . 6 (𝐻:𝑌𝐵𝐻 Fn 𝑌)
86, 7syl 17 . . . . 5 (𝜑𝐻 Fn 𝑌)
9 cvmlift2lem9a.4 . . . . 5 (𝜑𝑀𝑌)
10 fnssres 5964 . . . . 5 ((𝐻 Fn 𝑌𝑀𝑌) → (𝐻𝑀) Fn 𝑀)
118, 9, 10syl2anc 692 . . . 4 (𝜑 → (𝐻𝑀) Fn 𝑀)
12 df-ima 5089 . . . . 5 (𝐻𝑀) = ran (𝐻𝑀)
13 cvmlift2lem9a.6 . . . . 5 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1412, 13syl5eqssr 3631 . . . 4 (𝜑 → ran (𝐻𝑀) ⊆ 𝑊)
15 df-f 5853 . . . 4 ((𝐻𝑀):𝑀𝑊 ↔ ((𝐻𝑀) Fn 𝑀 ∧ ran (𝐻𝑀) ⊆ 𝑊))
1611, 14, 15sylanbrc 697 . . 3 (𝜑 → (𝐻𝑀):𝑀𝑊)
17 cvmlift2lem9a.2 . . . . . . . . . . 11 (𝜑𝑇 ∈ (𝑆𝐴))
18 cvmlift2lem9a.3 . . . . . . . . . . . 12 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
1918simpld 475 . . . . . . . . . . 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 30983 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → (𝐹𝑊):𝑊1-1-onto𝐴)
221, 17, 19, 21syl3anc 1323 . . . . . . . . . 10 (𝜑 → (𝐹𝑊):𝑊1-1-onto𝐴)
2322adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑊):𝑊1-1-onto𝐴)
24 f1of1 6095 . . . . . . . . 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 20647 . . . . . . . . . . 11 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
283, 27sylib 208 . . . . . . . . . 10 (𝜑𝐶 ∈ (TopOn‘𝐵))
2920cvmsss 30978 . . . . . . . . . . . . 13 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
3017, 29syl 17 . . . . . . . . . . . 12 (𝜑𝑇𝐶)
3130, 19sseldd 3585 . . . . . . . . . . 11 (𝜑𝑊𝐶)
32 toponss 20643 . . . . . . . . . . 11 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐶) → 𝑊𝐵)
3328, 31, 32syl2anc 692 . . . . . . . . . 10 (𝜑𝑊𝐵)
34 resttopon 20878 . . . . . . . . . 10 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐵) → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
3528, 33, 34syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
36 toponss 20643 . . . . . . . . 9 (((𝐶t 𝑊) ∈ (TopOn‘𝑊) ∧ 𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
3735, 36sylan 488 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
38 f1imacnv 6112 . . . . . . . 8 (((𝐹𝑊):𝑊1-1𝐴𝑥𝑊) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
3925, 37, 38syl2anc 692 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
4039imaeq2d 5427 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = ((𝐻𝑀) “ 𝑥))
41 imaco 5601 . . . . . . 7 (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)))
42 cnvco 5270 . . . . . . . . 9 ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐻𝑀) ∘ (𝐹𝑊))
43 cores 5599 . . . . . . . . . . . . 13 (ran (𝐻𝑀) ⊆ 𝑊 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
4414, 43syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
45 resco 5600 . . . . . . . . . . . 12 ((𝐹𝐻) ↾ 𝑀) = (𝐹 ∘ (𝐻𝑀))
4644, 45syl6eqr 2673 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4746adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4847cnveqd 5260 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4942, 48syl5eqr 2669 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) ∘ (𝐹𝑊)) = ((𝐹𝐻) ↾ 𝑀))
5049imaeq1d 5426 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5141, 50syl5eqr 2669 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5240, 51eqtr3d 2657 . . . . 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 21002 . . . . . . . 8 (((𝐹𝐻) ∈ (𝐾 Cn 𝐽) ∧ 𝑀𝑌) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5653, 9, 55syl2anc 692 . . . . . . 7 (𝜑 → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5756adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
58 resima2 5393 . . . . . . . 8 (𝑥𝑊 → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
5937, 58syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
601adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
61 restopn2 20894 . . . . . . . . . 10 ((𝐶 ∈ Top ∧ 𝑊𝐶) → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
623, 31, 61syl2anc 692 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
6362simprbda 652 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝐶)
64 cvmopn 30991 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝐶) → (𝐹𝑥) ∈ 𝐽)
6560, 63, 64syl2anc 692 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑥) ∈ 𝐽)
6659, 65eqeltrd 2698 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) ∈ 𝐽)
67 cnima 20982 . . . . . 6 ((((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽) ∧ ((𝐹𝑊) “ 𝑥) ∈ 𝐽) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6857, 66, 67syl2anc 692 . . . . 5 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6952, 68eqeltrd 2698 . . . 4 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
7069ralrimiva 2960 . . 3 (𝜑 → ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
71 cvmlift2lem9a.k . . . . . 6 (𝜑𝐾 ∈ Top)
7254toptopon 20647 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
7371, 72sylib 208 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
74 resttopon 20878 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
7573, 9, 74syl2anc 692 . . . 4 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
76 iscn 20952 . . . 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 692 . . 3 (𝜑 → ((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ↔ ((𝐻𝑀):𝑀𝑊 ∧ ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))))
7816, 70, 77mpbir2and 956 . 2 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)))
795, 78sseldd 3585 1 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3553  cin 3555  wss 3556  c0 3893  𝒫 cpw 4132  {csn 4150   cuni 4404  cmpt 4675  ccnv 5075  ran crn 5077  cres 5078  cima 5079  ccom 5080   Fn wfn 5844  wf 5845  1-1wf1 5846  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607  t crest 16005  Topctop 20620  TopOnctopon 20637   Cn ccn 20941  Homeochmeo 21469   CovMap ccvm 30966
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  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-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-oadd 7512  df-er 7690  df-map 7807  df-en 7903  df-fin 7906  df-fi 8264  df-rest 16007  df-topgen 16028  df-top 20621  df-topon 20638  df-bases 20664  df-cn 20944  df-hmeo 21471  df-cvm 30967
This theorem is referenced by:  cvmlift2lem9  31022  cvmlift3lem7  31036
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