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Theorem cnpresti 23296
Description: One direction of cnprest 23297 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Hypothesis
Ref Expression
cnprest.1 𝑋 = 𝐽
Assertion
Ref Expression
cnpresti ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))

Proof of Theorem cnpresti
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnprest.1 . . . . 5 𝑋 = 𝐽
2 eqid 2737 . . . . 5 𝐾 = 𝐾
31, 2cnpf 23255 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋 𝐾)
433ad2ant3 1136 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋 𝐾)
5 simp1 1137 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴𝑋)
64, 5fssresd 6775 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴):𝐴 𝐾)
7 simpl2 1193 . . . . . 6 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → 𝑃𝐴)
87fvresd 6926 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
98eleq1d 2826 . . . 4 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝑦))
10 cnpimaex 23264 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
11103expia 1122 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
12113ad2antl3 1188 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
13 idd 24 . . . . . . . . . . 11 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥𝑃𝑥))
14 simp2 1138 . . . . . . . . . . 11 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝐴)
1513, 14jctird 526 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥 → (𝑃𝑥𝑃𝐴)))
16 elin 3967 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
1715, 16imbitrrdi 252 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥𝑃 ∈ (𝑥𝐴)))
18 inss1 4237 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
19 imass2 6120 . . . . . . . . . . 11 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
2018, 19ax-mp 5 . . . . . . . . . 10 (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥)
21 id 22 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → (𝐹𝑥) ⊆ 𝑦)
2220, 21sstrid 3995 . . . . . . . . 9 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
2317, 22anim12d1 610 . . . . . . . 8 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
2423reximdv 3170 . . . . . . 7 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
25 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
2625inex1 5317 . . . . . . . . 9 (𝑥𝐴) ∈ V
2726a1i 11 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
28 cnptop1 23250 . . . . . . . . . 10 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
29283ad2ant3 1136 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3029uniexd 7762 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ V)
315, 1sseqtrdi 4024 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 𝐽)
3230, 31ssexd 5324 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ∈ V)
33 elrest 17472 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
3429, 32, 33syl2anc 584 . . . . . . . 8 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
35 simpr 484 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
3635eleq2d 2827 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
3735imaeq2d 6078 . . . . . . . . . . 11 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
38 inss2 4238 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
39 resima2 6034 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
4038, 39ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
4137, 40eqtrdi 2793 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
4241sseq1d 4015 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
4336, 42anbi12d 632 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
4427, 34, 43rexxfr2d 5411 . . . . . . 7 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
4524, 44sylibrd 259 . . . . . 6 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
4645adantr 480 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
4712, 46syld 47 . . . 4 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
489, 47sylbid 240 . . 3 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
4948ralrimiva 3146 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
501toptopon 22923 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5129, 50sylib 218 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
52 resttopon 23169 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
5351, 5, 52syl2anc 584 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
54 cnptop2 23251 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
55543ad2ant3 1136 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
562toptopon 22923 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
5755, 56sylib 218 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ (TopOn‘ 𝐾))
58 iscnp 23245 . . 3 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑃𝐴) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹𝐴):𝐴 𝐾 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
5953, 57, 14, 58syl3anc 1373 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹𝐴):𝐴 𝐾 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
606, 49, 59mpbir2and 713 1 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951   cuni 4907  cres 5687  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916   CnP ccnp 23233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-map 8868  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cnp 23236
This theorem is referenced by:  efrlim  27012  efrlimOLD  27013  cvmlift2lem11  35318
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