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Theorem flfcnp 23994
Description: A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
Assertion
Ref Expression
flfcnp (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺𝐹)))

Proof of Theorem flfcnp
StepHypRef Expression
1 simprl 769 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
2 flfval 23980 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
32adantr 479 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
41, 3eleqtrd 2828 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
5 simprr 771 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))
6 cnpflfi 23989 . . 3 ((𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐺𝐴) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺))
74, 5, 6syl2anc 582 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺𝐴) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺))
8 cnptop2 23233 . . . . . . . 8 (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
98ad2antll 727 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ Top)
10 toptopon2 22906 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
119, 10sylib 217 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ (TopOn‘ 𝐾))
12 toponmax 22914 . . . . . 6 (𝐾 ∈ (TopOn‘ 𝐾) → 𝐾𝐾)
1311, 12syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾𝐾)
14 simpl1 1188 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋))
15 toponmax 22914 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1614, 15syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝑋𝐽)
17 simpl2 1189 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐿 ∈ (Fil‘𝑌))
18 filfbas 23838 . . . . . 6 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
1917, 18syl 17 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐿 ∈ (fBas‘𝑌))
20 cnpf2 23240 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺:𝑋 𝐾)
2114, 11, 5, 20syl3anc 1368 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐺:𝑋 𝐾)
22 simpl3 1190 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐹:𝑌𝑋)
23 fmco 23951 . . . . 5 ((( 𝐾𝐾𝑋𝐽𝐿 ∈ (fBas‘𝑌)) ∧ (𝐺:𝑋 𝐾𝐹:𝑌𝑋)) → (( 𝐾 FilMap (𝐺𝐹))‘𝐿) = (( 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))
2413, 16, 19, 21, 22, 23syl32anc 1375 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (( 𝐾 FilMap (𝐺𝐹))‘𝐿) = (( 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))
2524oveq2d 7430 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐾 fLim (( 𝐾 FilMap (𝐺𝐹))‘𝐿)) = (𝐾 fLim (( 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿))))
26 fco 6742 . . . . 5 ((𝐺:𝑋 𝐾𝐹:𝑌𝑋) → (𝐺𝐹):𝑌 𝐾)
2721, 22, 26syl2anc 582 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺𝐹):𝑌 𝐾)
28 flfval 23980 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ (𝐺𝐹):𝑌 𝐾) → ((𝐾 fLimf 𝐿)‘(𝐺𝐹)) = (𝐾 fLim (( 𝐾 FilMap (𝐺𝐹))‘𝐿)))
2911, 17, 27, 28syl3anc 1368 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf 𝐿)‘(𝐺𝐹)) = (𝐾 fLim (( 𝐾 FilMap (𝐺𝐹))‘𝐿)))
30 fmfil 23934 . . . . 5 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
3116, 19, 22, 30syl3anc 1368 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
32 flfval 23980 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋) ∧ 𝐺:𝑋 𝐾) → ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺) = (𝐾 fLim (( 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿))))
3311, 31, 21, 32syl3anc 1368 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺) = (𝐾 fLim (( 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿))))
3425, 29, 333eqtr4d 2776 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf 𝐿)‘(𝐺𝐹)) = ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺))
357, 34eleqtrrd 2829 1 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099   cuni 4906  ccom 5677  wf 6540  cfv 6544  (class class class)co 7414  fBascfbas 21325  Topctop 22881  TopOnctopon 22898   CnP ccnp 23215  Filcfil 23835   FilMap cfm 23923   fLim cflim 23924   fLimf cflf 23925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7993  df-2nd 7994  df-map 8847  df-fbas 21334  df-fg 21335  df-top 22882  df-topon 22899  df-ntr 23010  df-nei 23088  df-cnp 23218  df-fil 23836  df-fm 23928  df-flim 23929  df-flf 23930
This theorem is referenced by:  flfcnp2  23997  tsmsmhm  24136
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