MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnpresti Structured version   Visualization version   GIF version

Theorem cnpresti 21011
Description: One direction of cnprest 21012 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 2621 . . . . 5 𝐾 = 𝐾
31, 2cnpf 20970 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋 𝐾)
433ad2ant3 1082 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋 𝐾)
5 simp1 1059 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴𝑋)
64, 5fssresd 6033 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴):𝐴 𝐾)
7 simpl2 1063 . . . . . 6 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → 𝑃𝐴)
8 fvres 6169 . . . . . 6 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
97, 8syl 17 . . . . 5 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
109eleq1d 2683 . . . 4 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝑦))
11 cnpimaex 20979 . . . . . . 7 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
12113expia 1264 . . . . . 6 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
13123ad2antl3 1223 . . . . 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 1060 . . . . . . . . . . 11 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝐴)
1614, 15jctird 566 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥 → (𝑃𝑥𝑃𝐴)))
17 elin 3779 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
1816, 17syl6ibr 242 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃𝑥𝑃 ∈ (𝑥𝐴)))
19 inss1 3816 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝑥
20 imass2 5465 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
2119, 20ax-mp 5 . . . . . . . . . . 11 (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥)
22 id 22 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐹𝑥) ⊆ 𝑦)
2321, 22syl5ss 3598 . . . . . . . . . 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 3011 . . . . . . 7 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
27 vex 3192 . . . . . . . . . 10 𝑥 ∈ V
2827inex1 4764 . . . . . . . . 9 (𝑥𝐴) ∈ V
2928a1i 11 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
30 cnptop1 20965 . . . . . . . . . 10 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
31303ad2ant3 1082 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
32 uniexg 6915 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝐽 ∈ V)
3331, 32syl 17 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ V)
345, 1syl6sseq 3635 . . . . . . . . . 10 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 𝐽)
3533, 34ssexd 4770 . . . . . . . . 9 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ∈ V)
36 elrest 16016 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
3731, 35, 36syl2anc 692 . . . . . . . 8 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
38 simpr 477 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
3938eleq2d 2684 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
4038imaeq2d 5430 . . . . . . . . . . 11 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
41 inss2 3817 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
42 resima2 5396 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
4341, 42ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
4440, 43syl6eq 2671 . . . . . . . . . 10 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
4544sseq1d 3616 . . . . . . . . 9 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
4639, 45anbi12d 746 . . . . . . . 8 (((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
4729, 37, 46rexxfr2d 4848 . . . . . . 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 481 . . . . 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 2961 . 2 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
531toptopon 20653 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5431, 53sylib 208 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
55 resttopon 20884 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
5654, 5, 55syl2anc 692 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
57 cnptop2 20966 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
58573ad2ant3 1082 . . . 4 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
592toptopon 20653 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
6058, 59sylib 208 . . 3 ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ (TopOn‘ 𝐾))
61 iscnp 20960 . . 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 1323 . 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 956 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  cin 3558  wss 3559   cuni 4407  cres 5081  cima 5082  wf 5848  cfv 5852  (class class class)co 6610  t crest 16009  Topctop 20626  TopOnctopon 20643   CnP ccnp 20948
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-fin 7910  df-fi 8268  df-rest 16011  df-topgen 16032  df-top 20627  df-topon 20644  df-bases 20670  df-cnp 20951
This theorem is referenced by:  efrlim  24609  cvmlift2lem11  31030
  Copyright terms: Public domain W3C validator