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Mirrors > Home > MPE Home > Th. List > suppss | Structured version Visualization version Unicode version |
Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
suppss.f | |
suppss.n |
Ref | Expression |
---|---|
suppss | supp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppss.f | . . . . . . . 8 | |
2 | ffn 6045 | . . . . . . . 8 | |
3 | 1, 2 | syl 17 | . . . . . . 7 |
4 | 3 | adantl 482 | . . . . . 6 |
5 | fdm 6051 | . . . . . . . 8 | |
6 | dmexg 7097 | . . . . . . . . . 10 | |
7 | 6 | adantr 481 | . . . . . . . . 9 |
8 | eleq1 2689 | . . . . . . . . . 10 | |
9 | 8 | eqcoms 2630 | . . . . . . . . 9 |
10 | 7, 9 | syl5ibr 236 | . . . . . . . 8 |
11 | 1, 5, 10 | 3syl 18 | . . . . . . 7 |
12 | 11 | impcom 446 | . . . . . 6 |
13 | simplr 792 | . . . . . 6 | |
14 | elsuppfn 7303 | . . . . . 6 supp | |
15 | 4, 12, 13, 14 | syl3anc 1326 | . . . . 5 supp |
16 | eldif 3584 | . . . . . . . . 9 | |
17 | suppss.n | . . . . . . . . . 10 | |
18 | 17 | adantll 750 | . . . . . . . . 9 |
19 | 16, 18 | sylan2br 493 | . . . . . . . 8 |
20 | 19 | expr 643 | . . . . . . 7 |
21 | 20 | necon1ad 2811 | . . . . . 6 |
22 | 21 | expimpd 629 | . . . . 5 |
23 | 15, 22 | sylbid 230 | . . . 4 supp |
24 | 23 | ssrdv 3609 | . . 3 supp |
25 | 24 | ex 450 | . 2 supp |
26 | supp0prc 7298 | . . . 4 supp | |
27 | 0ss 3972 | . . . 4 | |
28 | 26, 27 | syl6eqss 3655 | . . 3 supp |
29 | 28 | a1d 25 | . 2 supp |
30 | 25, 29 | pm2.61i 176 | 1 supp |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 cdif 3571 wss 3574 c0 3915 cdm 5114 wfn 5883 wf 5884 cfv 5888 (class class class)co 6650 supp csupp 7295 |
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-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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-supp 7296 |
This theorem is referenced by: fsuppco2 8308 fsuppcor 8309 cantnfp1lem1 8575 cantnfp1lem3 8577 gsumzaddlem 18321 gsumzmhm 18337 gsum2d2lem 18372 lcomfsupp 18903 psrbaglesupp 19368 mplsubglem 19434 mpllsslem 19435 mplsubrglem 19439 mvrcl 19449 evlslem3 19514 frlmssuvc1 20133 frlmsslsp 20135 frlmup2 20138 deg1mul3le 23876 jensen 24715 resf1o 29505 |
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