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Theorem cvmlift2lem9a 31616
Description: Lemma for cvmlift2 31629 and cvmlift3 31641. (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 31573 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐶 ∈ Top)
4 cnrest2r 21305 . . 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 6183 . . . . . 6 (𝐻:𝑌𝐵𝐻 Fn 𝑌)
86, 7syl 17 . . . . 5 (𝜑𝐻 Fn 𝑌)
9 cvmlift2lem9a.4 . . . . 5 (𝜑𝑀𝑌)
10 fnssres 6142 . . . . 5 ((𝐻 Fn 𝑌𝑀𝑌) → (𝐻𝑀) Fn 𝑀)
118, 9, 10syl2anc 573 . . . 4 (𝜑 → (𝐻𝑀) Fn 𝑀)
12 df-ima 5262 . . . . 5 (𝐻𝑀) = ran (𝐻𝑀)
13 cvmlift2lem9a.6 . . . . 5 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1412, 13syl5eqssr 3799 . . . 4 (𝜑 → ran (𝐻𝑀) ⊆ 𝑊)
15 df-f 6033 . . . 4 ((𝐻𝑀):𝑀𝑊 ↔ ((𝐻𝑀) Fn 𝑀 ∧ ran (𝐻𝑀) ⊆ 𝑊))
1611, 14, 15sylanbrc 572 . . 3 (𝜑 → (𝐻𝑀):𝑀𝑊)
17 cvmlift2lem9a.2 . . . . . . . . . . 11 (𝜑𝑇 ∈ (𝑆𝐴))
18 cvmlift2lem9a.3 . . . . . . . . . . . 12 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
1918simpld 482 . . . . . . . . . . 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 31585 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → (𝐹𝑊):𝑊1-1-onto𝐴)
221, 17, 19, 21syl3anc 1476 . . . . . . . . . 10 (𝜑 → (𝐹𝑊):𝑊1-1-onto𝐴)
2322adantr 466 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑊):𝑊1-1-onto𝐴)
24 f1of1 6275 . . . . . . . . 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 20935 . . . . . . . . . . 11 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
283, 27sylib 208 . . . . . . . . . 10 (𝜑𝐶 ∈ (TopOn‘𝐵))
2920cvmsss 31580 . . . . . . . . . . . . 13 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
3017, 29syl 17 . . . . . . . . . . . 12 (𝜑𝑇𝐶)
3130, 19sseldd 3753 . . . . . . . . . . 11 (𝜑𝑊𝐶)
32 toponss 20945 . . . . . . . . . . 11 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐶) → 𝑊𝐵)
3328, 31, 32syl2anc 573 . . . . . . . . . 10 (𝜑𝑊𝐵)
34 resttopon 21179 . . . . . . . . . 10 ((𝐶 ∈ (TopOn‘𝐵) ∧ 𝑊𝐵) → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
3528, 33, 34syl2anc 573 . . . . . . . . 9 (𝜑 → (𝐶t 𝑊) ∈ (TopOn‘𝑊))
36 toponss 20945 . . . . . . . . 9 (((𝐶t 𝑊) ∈ (TopOn‘𝑊) ∧ 𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
3735, 36sylan 569 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝑊)
38 f1imacnv 6292 . . . . . . . 8 (((𝐹𝑊):𝑊1-1𝐴𝑥𝑊) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
3925, 37, 38syl2anc 573 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)) = 𝑥)
4039imaeq2d 5605 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = ((𝐻𝑀) “ 𝑥))
41 imaco 5782 . . . . . . 7 (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥)))
42 cnvco 5444 . . . . . . . . 9 ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐻𝑀) ∘ (𝐹𝑊))
43 cores 5780 . . . . . . . . . . . . 13 (ran (𝐻𝑀) ⊆ 𝑊 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
4414, 43syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = (𝐹 ∘ (𝐻𝑀)))
45 resco 5781 . . . . . . . . . . . 12 ((𝐹𝐻) ↾ 𝑀) = (𝐹 ∘ (𝐻𝑀))
4644, 45syl6eqr 2823 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4746adantr 466 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4847cnveqd 5434 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) ∘ (𝐻𝑀)) = ((𝐹𝐻) ↾ 𝑀))
4942, 48syl5eqr 2819 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) ∘ (𝐹𝑊)) = ((𝐹𝐻) ↾ 𝑀))
5049imaeq1d 5604 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐻𝑀) ∘ (𝐹𝑊)) “ ((𝐹𝑊) “ 𝑥)) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5141, 50syl5eqr 2819 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ ((𝐹𝑊) “ ((𝐹𝑊) “ 𝑥))) = (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)))
5240, 51eqtr3d 2807 . . . . 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 21303 . . . . . . . 8 (((𝐹𝐻) ∈ (𝐾 Cn 𝐽) ∧ 𝑀𝑌) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5653, 9, 55syl2anc 573 . . . . . . 7 (𝜑 → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
5756adantr 466 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽))
58 resima2 5571 . . . . . . . 8 (𝑥𝑊 → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
5937, 58syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) = (𝐹𝑥))
601adantr 466 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
61 restopn2 21195 . . . . . . . . . 10 ((𝐶 ∈ Top ∧ 𝑊𝐶) → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
623, 31, 61syl2anc 573 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐶t 𝑊) ↔ (𝑥𝐶𝑥𝑊)))
6362simprbda 486 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → 𝑥𝐶)
64 cvmopn 31593 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥𝐶) → (𝐹𝑥) ∈ 𝐽)
6560, 63, 64syl2anc 573 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (𝐹𝑥) ∈ 𝐽)
6659, 65eqeltrd 2850 . . . . . 6 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐹𝑊) “ 𝑥) ∈ 𝐽)
67 cnima 21283 . . . . . 6 ((((𝐹𝐻) ↾ 𝑀) ∈ ((𝐾t 𝑀) Cn 𝐽) ∧ ((𝐹𝑊) “ 𝑥) ∈ 𝐽) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6857, 66, 67syl2anc 573 . . . . 5 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → (((𝐹𝐻) ↾ 𝑀) “ ((𝐹𝑊) “ 𝑥)) ∈ (𝐾t 𝑀))
6952, 68eqeltrd 2850 . . . 4 ((𝜑𝑥 ∈ (𝐶t 𝑊)) → ((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
7069ralrimiva 3115 . . 3 (𝜑 → ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))
71 cvmlift2lem9a.k . . . . . 6 (𝜑𝐾 ∈ Top)
7254toptopon 20935 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
7371, 72sylib 208 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
74 resttopon 21179 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
7573, 9, 74syl2anc 573 . . . 4 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
76 iscn 21253 . . . 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 573 . . 3 (𝜑 → ((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)) ↔ ((𝐻𝑀):𝑀𝑊 ∧ ∀𝑥 ∈ (𝐶t 𝑊)((𝐻𝑀) “ 𝑥) ∈ (𝐾t 𝑀))))
7816, 70, 77mpbir2and 692 . 2 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn (𝐶t 𝑊)))
795, 78sseldd 3753 1 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cdif 3720  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   cuni 4574  cmpt 4863  ccnv 5248  ran crn 5250  cres 5251  cima 5252  ccom 5253   Fn wfn 6024  wf 6025  1-1wf1 6026  1-1-ontowf1o 6028  cfv 6029  (class class class)co 6791  t crest 16282  Topctop 20911  TopOnctopon 20928   Cn ccn 21242  Homeochmeo 21770   CovMap ccvm 31568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-oadd 7715  df-er 7894  df-map 8009  df-en 8108  df-fin 8111  df-fi 8471  df-rest 16284  df-topgen 16305  df-top 20912  df-topon 20929  df-bases 20964  df-cn 21245  df-hmeo 21772  df-cvm 31569
This theorem is referenced by:  cvmlift2lem9  31624  cvmlift3lem7  31638
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