Theorem flfcnp 23982

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 771	. . . 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 23968	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3	2	adantr 480	. . . 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 2839	. . 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 773	. . 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 23977	. . 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 585	. 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 23221	. . . . . . . 8 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
9	8	ad2antll 730	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ Top)
10		toptopon2 22896	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
11	9, 10	sylib 218	. . . . . 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 22904	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) → ∪ 𝐾 ∈ 𝐾)
13	11, 12	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ∪ 𝐾 ∈ 𝐾)
14		simpl1 1193	. . . . . 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 22904	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
16	14, 15	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝑋 ∈ 𝐽)
17		simpl2 1194	. . . . . 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 23826	. . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
19	17, 18	syl 17	. . . . 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 23228	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺:𝑋⟶∪ 𝐾)
21	14, 11, 5, 20	syl3anc 1374	. . . . 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 1195	. . . . 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 23939	. . . . 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 1381	. . . 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 7377	. . 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 6687	. . . . 5 ⊢ ((𝐺:𝑋⟶∪ 𝐾 ∧ 𝐹:𝑌⟶𝑋) → (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾)
27	21, 22, 26	syl2anc 585	. . . 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 23968	. . . 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 1374	. . 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 23922	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
31	16, 19, 22, 30	syl3anc 1374	. . . 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 23968	. . . 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 1374	. . 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 2782	. 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 2840	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∪ cuni 4851   ∘ ccom 5629  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  fBascfbas 21335  Topctop 22871  TopOnctopon 22888   CnP ccnp 23203  Filcfil 23823   FilMap cfm 23911   fLim cflim 23912   fLimf cflf 23913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8769  df-fbas 21344  df-fg 21345  df-top 22872  df-topon 22889  df-ntr 22998  df-nei 23076  df-cnp 23206  df-fil 23824  df-fm 23916  df-flim 23917  df-flf 23918

This theorem is referenced by:  flfcnp2  23985  tsmsmhm  24124