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Mirrors > Home > MPE Home > Th. List > flfval | Structured version Visualization version GIF version |
Description: Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
flfval | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponmax 20932 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
2 | filtop 21860 | . . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿) | |
3 | elmapg 8036 | . . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐿) → (𝐹 ∈ (𝑋 ↑𝑚 𝑌) ↔ 𝐹:𝑌⟶𝑋)) | |
4 | 1, 2, 3 | syl2an 495 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋 ↑𝑚 𝑌) ↔ 𝐹:𝑌⟶𝑋)) |
5 | 4 | biimpar 503 | . . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ (𝑋 ↑𝑚 𝑌)) |
6 | flffval 21994 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))) | |
7 | 6 | fveq1d 6354 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = ((𝑓 ∈ (𝑋 ↑𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹)) |
8 | oveq2 6821 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑋 FilMap 𝑓) = (𝑋 FilMap 𝐹)) | |
9 | 8 | fveq1d 6354 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑋 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿)) |
10 | 9 | oveq2d 6829 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
11 | eqid 2760 | . . . . 5 ⊢ (𝑓 ∈ (𝑋 ↑𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋 ↑𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) | |
12 | ovex 6841 | . . . . 5 ⊢ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V | |
13 | 10, 11, 12 | fvmpt 6444 | . . . 4 ⊢ (𝐹 ∈ (𝑋 ↑𝑚 𝑌) → ((𝑓 ∈ (𝑋 ↑𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
14 | 7, 13 | sylan9eq 2814 | . . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹 ∈ (𝑋 ↑𝑚 𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
15 | 5, 14 | syldan 488 | . 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
16 | 15 | 3impa 1101 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ↦ cmpt 4881 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↑𝑚 cmap 8023 TopOnctopon 20917 Filcfil 21850 FilMap cfm 21938 fLim cflim 21939 fLimf cflf 21940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-map 8025 df-fbas 19945 df-top 20901 df-topon 20918 df-fil 21851 df-flf 21945 |
This theorem is referenced by: flfnei 21996 isflf 21998 hausflf 22002 flfcnp 22009 flfssfcf 22043 uffcfflf 22044 cnpfcf 22046 cnextcn 22072 tsmscls 22142 cnextucn 22308 cmetcaulem 23286 fmcncfil 30286 |
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