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Theorem cnpresti 23311
Description: One direction of cnprest 23312 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 2734 . . . . 5 𝐾 = 𝐾
31, 2cnpf 23270 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋 𝐾)
433ad2ant3 1134 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋 𝐾)
5 simp1 1135 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴𝑋)
64, 5fssresd 6775 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴):𝐴 𝐾)
7 simpl2 1191 . . . . . 6 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → 𝑃𝐴)
87fvresd 6926 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
98eleq1d 2823 . . . 4 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝑦))
10 cnpimaex 23279 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
11103expia 1120 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
12113ad2antl3 1186 . . . . 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 1136 . . . . . . . . . . 11 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝐴)
1513, 14jctird 526 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥 → (𝑃𝑥𝑃𝐴)))
16 elin 3978 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
1715, 16imbitrrdi 252 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥𝑃 ∈ (𝑥𝐴)))
18 inss1 4244 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
19 imass2 6122 . . . . . . . . . . 11 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
2018, 19ax-mp 5 . . . . . . . . . 10 (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥)
21 id 22 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → (𝐹𝑥) ⊆ 𝑦)
2220, 21sstrid 4006 . . . . . . . . 9 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
2317, 22anim12d1 610 . . . . . . . 8 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
2423reximdv 3167 . . . . . . 7 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
25 vex 3481 . . . . . . . . . 10 𝑥 ∈ V
2625inex1 5322 . . . . . . . . 9 (𝑥𝐴) ∈ V
2726a1i 11 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
28 cnptop1 23265 . . . . . . . . . 10 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
29283ad2ant3 1134 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
3029uniexd 7760 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ V)
315, 1sseqtrdi 4045 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 𝐽)
3230, 31ssexd 5329 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ∈ V)
33 elrest 17473 . . . . . . . . 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 2824 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
3735imaeq2d 6079 . . . . . . . . . . 11 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
38 inss2 4245 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
39 resima2 6035 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
4038, 39ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
4137, 40eqtrdi 2790 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
4241sseq1d 4026 . . . . . . . . 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 5416 . . . . . . 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 3143 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
501toptopon 22938 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5129, 50sylib 218 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
52 resttopon 23184 . . . 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 23266 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
55543ad2ant3 1134 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
562toptopon 22938 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
5755, 56sylib 218 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ (TopOn‘ 𝐾))
58 iscnp 23260 . . 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 1370 . 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 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cin 3961  wss 3962   cuni 4911  cres 5690  cima 5691  wf 6558  cfv 6562  (class class class)co 7430  t crest 17466  Topctop 22914  TopOnctopon 22931   CnP ccnp 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-map 8866  df-en 8984  df-fin 8987  df-fi 9448  df-rest 17468  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cnp 23251
This theorem is referenced by:  efrlim  27026  efrlimOLD  27027  cvmlift2lem11  35297
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