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Mirrors > Home > HSE Home > Th. List > hlimf | Structured version Visualization version GIF version |
Description: Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimf | β’ βπ£ :dom βπ£ βΆ β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . . 7 β’ β¨β¨ +β , Β·β β©, normββ© = β¨β¨ +β , Β·β β©, normββ© | |
2 | eqid 2733 | . . . . . . 7 β’ (IndMetββ¨β¨ +β , Β·β β©, normββ©) = (IndMetββ¨β¨ +β , Β·β β©, normββ©) | |
3 | 1, 2 | hhxmet 30166 | . . . . . 6 β’ (IndMetββ¨β¨ +β , Β·β β©, normββ©) β (βMetβ β) |
4 | eqid 2733 | . . . . . . 7 β’ (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) = (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) | |
5 | 4 | methaus 23899 | . . . . . 6 β’ ((IndMetββ¨β¨ +β , Β·β β©, normββ©) β (βMetβ β) β (MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) β Haus) |
6 | lmfun 22755 | . . . . . 6 β’ ((MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)) β Haus β Fun (βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©)))) | |
7 | 3, 5, 6 | mp2b 10 | . . . . 5 β’ Fun (βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) |
8 | funres 6547 | . . . . 5 β’ (Fun (βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) β Fun ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β))) | |
9 | 7, 8 | ax-mp 5 | . . . 4 β’ Fun ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β)) |
10 | 1, 2, 4 | hhlm 30190 | . . . . 5 β’ βπ£ = ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β)) |
11 | 10 | funeqi 6526 | . . . 4 β’ (Fun βπ£ β Fun ((βπ‘β(MetOpenβ(IndMetββ¨β¨ +β , Β·β β©, normββ©))) βΎ ( β βm β))) |
12 | 9, 11 | mpbir 230 | . . 3 β’ Fun βπ£ |
13 | funfn 6535 | . . 3 β’ (Fun βπ£ β βπ£ Fn dom βπ£ ) | |
14 | 12, 13 | mpbi 229 | . 2 β’ βπ£ Fn dom βπ£ |
15 | funfvbrb 7005 | . . . . 5 β’ (Fun βπ£ β (π₯ β dom βπ£ β π₯ βπ£ ( βπ£ βπ₯))) | |
16 | 12, 15 | ax-mp 5 | . . . 4 β’ (π₯ β dom βπ£ β π₯ βπ£ ( βπ£ βπ₯)) |
17 | fvex 6859 | . . . . 5 β’ ( βπ£ βπ₯) β V | |
18 | 17 | hlimveci 30181 | . . . 4 β’ (π₯ βπ£ ( βπ£ βπ₯) β ( βπ£ βπ₯) β β) |
19 | 16, 18 | sylbi 216 | . . 3 β’ (π₯ β dom βπ£ β ( βπ£ βπ₯) β β) |
20 | 19 | rgen 3063 | . 2 β’ βπ₯ β dom βπ£ ( βπ£ βπ₯) β β |
21 | ffnfv 7070 | . 2 β’ ( βπ£ :dom βπ£ βΆ β β ( βπ£ Fn dom βπ£ β§ βπ₯ β dom βπ£ ( βπ£ βπ₯) β β)) | |
22 | 14, 20, 21 | mpbir2an 710 | 1 β’ βπ£ :dom βπ£ βΆ β |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β wcel 2107 βwral 3061 β¨cop 4596 class class class wbr 5109 dom cdm 5637 βΎ cres 5639 Fun wfun 6494 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 βm cmap 8771 βcn 12161 βMetcxmet 20804 MetOpencmopn 20809 βπ‘clm 22600 Hauscha 22682 IndMetcims 29582 βchba 29910 +β cva 29911 Β·β csm 29912 normβcno 29914 βπ£ chli 29918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 ax-hilex 29990 ax-hfvadd 29991 ax-hvcom 29992 ax-hvass 29993 ax-hv0cl 29994 ax-hvaddid 29995 ax-hfvmul 29996 ax-hvmulid 29997 ax-hvmulass 29998 ax-hvdistr1 29999 ax-hvdistr2 30000 ax-hvmul0 30001 ax-hfi 30070 ax-his1 30073 ax-his2 30074 ax-his3 30075 ax-his4 30076 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-icc 13280 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-topgen 17333 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-bases 22319 df-lm 22603 df-haus 22689 df-grpo 29484 df-gid 29485 df-ginv 29486 df-gdiv 29487 df-ablo 29536 df-vc 29550 df-nv 29583 df-va 29586 df-ba 29587 df-sm 29588 df-0v 29589 df-vs 29590 df-nmcv 29591 df-ims 29592 df-hnorm 29959 df-hvsub 29962 df-hlim 29963 |
This theorem is referenced by: hlimuni 30229 hhsscms 30269 occllem 30294 occl 30295 chscllem2 30629 chscllem4 30631 |
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