Theorem flfcnp 24129

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:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)))

Proof of Theorem flfcnp

Step	Hyp	Ref	Expression
1		simprl 782	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
2		flfval 24115	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3	2	adantr 485	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
4	1, 3	eleqtrd 2871	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
5		simprr 784	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))
6		cnpflfi 24124	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐺‘𝐴) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺))
7	4, 5, 6	syl2anc 595	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺))
8		cnptop2 23368	. . . . . . . 8 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
9	8	ad2antll 741	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ Top)
10		toptopon2 23043	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
11	9, 10	sylib 221	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
12		toponmax 23051	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → ∪ 𝐾 ∈ 𝐾)
13	11, 12	syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ∪ 𝐾 ∈ 𝐾)
14		simpl1 1208	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋))
15		toponmax 23051	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
16	14, 15	syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝑋 ∈ 𝐽)
17		simpl2 1209	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐿 ∈ (Fil‘𝑌))
18		filfbas 23973	. . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
19	17, 18	syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐿 ∈ (fBas‘𝑌))
20		cnpf2 23375	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺:𝑋⟶∪ 𝐾)
21	14, 11, 5, 20	syl3anc 1396	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐺:𝑋⟶∪ 𝐾)
22		simpl3 1210	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐹:𝑌⟶𝑋)
23		fmco 24086	. . . . 5 ⊢ (((∪ 𝐾 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌)) ∧ (𝐺:𝑋⟶∪ 𝐾 ∧ 𝐹:𝑌⟶𝑋)) → ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿) = ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))
24	13, 16, 19, 21, 22, 23	syl32anc 1403	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿) = ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))
25	24	oveq2d 7427	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐾 fLim ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿)) = (𝐾 fLim ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿))))
26		fco 6731	. . . . 5 ⊢ ((𝐺:𝑋⟶∪ 𝐾 ∧ 𝐹:𝑌⟶𝑋) → (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾)
27	21, 22, 26	syl2anc 595	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾)
28		flfval 24115	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾) → ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)) = (𝐾 fLim ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿)))
29	11, 17, 27, 28	syl3anc 1396	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)) = (𝐾 fLim ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿)))
30		fmfil 24069	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
31	16, 19, 22, 30	syl3anc 1396	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
32		flfval 24115	. . . 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))))
33	11, 31, 21, 32	syl3anc 1396	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (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))))
34	25, 29, 33	3eqtr4d 2814	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)) = ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺))
35	7, 34	eleqtrrd 2872	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∪ cuni 4876   ∘ ccom 5666  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  fBascfbas 21478  Topctop 23018  TopOnctopon 23035   CnP ccnp 23350  Filcfil 23970   FilMap cfm 24058   fLim cflim 24059   fLimf cflf 24060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-map 8825  df-fbas 21487  df-fg 21488  df-top 23019  df-topon 23036  df-ntr 23145  df-nei 23223  df-cnp 23353  df-fil 23971  df-fm 24063  df-flim 24064  df-flf 24065

This theorem is referenced by:  flfcnp2  24132  tsmsmhm  24271