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Mirrors > Home > MPE Home > Th. List > conncompcld | Structured version Visualization version Unicode version |
Description: The connected component containing is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
conncomp.2 | ↾t Conn |
Ref | Expression |
---|---|
conncompcld | TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 20718 | . . . . . 6 TopOn | |
2 | 1 | adantr 481 | . . . . 5 TopOn |
3 | conncomp.2 | . . . . . . 7 ↾t Conn | |
4 | ssrab2 3687 | . . . . . . . 8 ↾t Conn | |
5 | sspwuni 4611 | . . . . . . . 8 ↾t Conn ↾t Conn | |
6 | 4, 5 | mpbi 220 | . . . . . . 7 ↾t Conn |
7 | 3, 6 | eqsstri 3635 | . . . . . 6 |
8 | toponuni 20719 | . . . . . . 7 TopOn | |
9 | 8 | adantr 481 | . . . . . 6 TopOn |
10 | 7, 9 | syl5sseq 3653 | . . . . 5 TopOn |
11 | eqid 2622 | . . . . . 6 | |
12 | 11 | clsss3 20863 | . . . . 5 |
13 | 2, 10, 12 | syl2anc 693 | . . . 4 TopOn |
14 | 13, 9 | sseqtr4d 3642 | . . 3 TopOn |
15 | 11 | sscls 20860 | . . . . 5 |
16 | 2, 10, 15 | syl2anc 693 | . . . 4 TopOn |
17 | 3 | conncompid 21234 | . . . 4 TopOn |
18 | 16, 17 | sseldd 3604 | . . 3 TopOn |
19 | simpl 473 | . . . 4 TopOn TopOn | |
20 | 7 | a1i 11 | . . . 4 TopOn |
21 | 3 | conncompconn 21235 | . . . 4 TopOn ↾t Conn |
22 | clsconn 21233 | . . . 4 TopOn ↾t Conn ↾t Conn | |
23 | 19, 20, 21, 22 | syl3anc 1326 | . . 3 TopOn ↾t Conn |
24 | 3 | conncompss 21236 | . . 3 ↾t Conn |
25 | 14, 18, 23, 24 | syl3anc 1326 | . 2 TopOn |
26 | 11 | iscld4 20869 | . . 3 |
27 | 2, 10, 26 | syl2anc 693 | . 2 TopOn |
28 | 25, 27 | mpbird 247 | 1 TopOn |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 crab 2916 wss 3574 cpw 4158 cuni 4436 cfv 5888 (class class class)co 6650 ↾t crest 16081 ctop 20698 TopOnctopon 20715 ccld 20820 ccl 20822 Conncconn 21214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-conn 21215 |
This theorem is referenced by: conncompclo 21238 |
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