Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rncmp | Structured version Visualization version Unicode version |
Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
rncmp | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . 2 | |
2 | eqid 2622 | . . . . . . 7 | |
3 | eqid 2622 | . . . . . . 7 | |
4 | 2, 3 | cnf 21050 | . . . . . 6 |
5 | 4 | adantl 482 | . . . . 5 |
6 | ffn 6045 | . . . . 5 | |
7 | 5, 6 | syl 17 | . . . 4 |
8 | dffn4 6121 | . . . 4 | |
9 | 7, 8 | sylib 208 | . . 3 |
10 | cntop2 21045 | . . . . . 6 | |
11 | 10 | adantl 482 | . . . . 5 |
12 | frn 6053 | . . . . . 6 | |
13 | 5, 12 | syl 17 | . . . . 5 |
14 | 3 | restuni 20966 | . . . . 5 ↾t |
15 | 11, 13, 14 | syl2anc 693 | . . . 4 ↾t |
16 | foeq3 6113 | . . . 4 ↾t ↾t | |
17 | 15, 16 | syl 17 | . . 3 ↾t |
18 | 9, 17 | mpbid 222 | . 2 ↾t |
19 | simpr 477 | . . 3 | |
20 | 3 | toptopon 20722 | . . . . 5 TopOn |
21 | 11, 20 | sylib 208 | . . . 4 TopOn |
22 | ssid 3624 | . . . . 5 | |
23 | 22 | a1i 11 | . . . 4 |
24 | cnrest2 21090 | . . . 4 TopOn ↾t | |
25 | 21, 23, 13, 24 | syl3anc 1326 | . . 3 ↾t |
26 | 19, 25 | mpbid 222 | . 2 ↾t |
27 | eqid 2622 | . . 3 ↾t ↾t | |
28 | 27 | cncmp 21195 | . 2 ↾t ↾t ↾t |
29 | 1, 18, 26, 28 | syl3anc 1326 | 1 ↾t |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wss 3574 cuni 4436 crn 5115 wfn 5883 wf 5884 wfo 5886 cfv 5888 (class class class)co 6650 ↾t crest 16081 ctop 20698 TopOnctopon 20715 ccn 21028 ccmp 21189 |
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-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-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-cmp 21190 |
This theorem is referenced by: imacmp 21200 kgencn2 21360 bndth 22757 |
Copyright terms: Public domain | W3C validator |