Theorem flfcnp 24332

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 783	. . . 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 24318	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
3	2	adantr 486	. . . 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 2862	. . 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 785	. . 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 24327	. . 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 596	. 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 23570	. . . . . . . 8 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top)
9	8	ad2antll 742	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ Top)
10		toptopon2 23245	. . . . . . 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 23253	. . . . . 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 1210	. . . . . 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 23253	. . . . . 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 1211	. . . . . 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 24176	. . . . . 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 23577	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺:𝑋⟶∪ 𝐾)
21	14, 11, 5, 20	syl3anc 1398	. . . . 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 1212	. . . . 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 24289	. . . . 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 1405	. . . 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 7440	. . 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 6739	. . . . 5 ⊢ ((𝐺:𝑋⟶∪ 𝐾 ∧ 𝐹:𝑌⟶𝑋) → (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾)
27	21, 22, 26	syl2anc 596	. . . 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 24318	. . . 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 1398	. . 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 24272	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
31	16, 19, 22, 30	syl3anc 1398	. . . 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 24318	. . . 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 1398	. . 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 2805	. 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 2863	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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∪ cuni 4866   ∘ ccom 5655  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  fBascfbas 21675  Topctop 23220  TopOnctopon 23237   CnP ccnp 23552  Filcfil 24173   FilMap cfm 24261   fLim cflim 24262   fLimf cflf 24263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-map 8856  df-fbas 21684  df-fg 21685  df-top 23221  df-topon 23238  df-ntr 23347  df-nei 23425  df-cnp 23555  df-fil 24174  df-fm 24266  df-flim 24267  df-flf 24268

This theorem is used by:  flfcnp2  24335  tsmsmhm  24474