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Theorem cnpresti 21140
Description: One direction of cnprest 21141 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 2651 . . . . 5 𝐾 = 𝐾
31, 2cnpf 21099 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋 𝐾)
433ad2ant3 1104 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋 𝐾)
5 simp1 1081 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴𝑋)
64, 5fssresd 6109 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴):𝐴 𝐾)
7 simpl2 1085 . . . . . 6 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → 𝑃𝐴)
8 fvres 6245 . . . . . 6 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
97, 8syl 17 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
109eleq1d 2715 . . . 4 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝑦))
11 cnpimaex 21108 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
12113expia 1286 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
13123ad2antl3 1245 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
14 idd 24 . . . . . . . . . . 11 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥𝑃𝑥))
15 simp2 1082 . . . . . . . . . . 11 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝐴)
1614, 15jctird 566 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥 → (𝑃𝑥𝑃𝐴)))
17 elin 3829 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
1816, 17syl6ibr 242 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥𝑃 ∈ (𝑥𝐴)))
19 inss1 3866 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝑥
20 imass2 5536 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
2119, 20ax-mp 5 . . . . . . . . . . 11 (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥)
22 id 22 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐹𝑥) ⊆ 𝑦)
2321, 22syl5ss 3647 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
2423a1i 11 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
2518, 24anim12d 585 . . . . . . . 8 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
2625reximdv 3045 . . . . . . 7 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
27 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
2827inex1 4832 . . . . . . . . 9 (𝑥𝐴) ∈ V
2928a1i 11 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
30 cnptop1 21094 . . . . . . . . . 10 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
31303ad2ant3 1104 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
32 uniexg 6997 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝐽 ∈ V)
3331, 32syl 17 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ V)
345, 1syl6sseq 3684 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 𝐽)
3533, 34ssexd 4838 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ∈ V)
36 elrest 16135 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
3731, 35, 36syl2anc 694 . . . . . . . 8 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
38 simpr 476 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
3938eleq2d 2716 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
4038imaeq2d 5501 . . . . . . . . . . 11 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
41 inss2 3867 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
42 resima2 5467 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
4341, 42ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
4440, 43syl6eq 2701 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
4544sseq1d 3665 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
4639, 45anbi12d 747 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
4729, 37, 46rexxfr2d 4913 . . . . . . 7 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
4826, 47sylibrd 249 . . . . . 6 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
4948adantr 480 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
5013, 49syld 47 . . . 4 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
5110, 50sylbid 230 . . 3 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
5251ralrimiva 2995 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
531toptopon 20770 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5431, 53sylib 208 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
55 resttopon 21013 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
5654, 5, 55syl2anc 694 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
57 cnptop2 21095 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
58573ad2ant3 1104 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
592toptopon 20770 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
6058, 59sylib 208 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ (TopOn‘ 𝐾))
61 iscnp 21089 . . 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)))))
6256, 60, 15, 61syl3anc 1366 . 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)))))
636, 52, 62mpbir2and 977 1 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607   cuni 4468  cres 5145  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  TopOnctopon 20763   CnP ccnp 21077
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-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-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-cnp 21080
This theorem is referenced by:  efrlim  24741  cvmlift2lem11  31421
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