Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fnsuppeq0 | Structured version Visualization version Unicode version |
Description: The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
fnsuppeq0 | supp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss0b 3973 | . . 3 supp supp | |
2 | un0 3967 | . . . . . . . 8 | |
3 | uncom 3757 | . . . . . . . 8 | |
4 | 2, 3 | eqtr3i 2646 | . . . . . . 7 |
5 | 4 | fneq2i 5986 | . . . . . 6 |
6 | 5 | biimpi 206 | . . . . 5 |
7 | 6 | 3ad2ant1 1082 | . . . 4 |
8 | fnex 6481 | . . . . 5 | |
9 | 8 | 3adant3 1081 | . . . 4 |
10 | simp3 1063 | . . . 4 | |
11 | 0in 3969 | . . . . 5 | |
12 | 11 | a1i 11 | . . . 4 |
13 | fnsuppres 7322 | . . . 4 supp | |
14 | 7, 9, 10, 12, 13 | syl121anc 1331 | . . 3 supp |
15 | 1, 14 | syl5bbr 274 | . 2 supp |
16 | fnresdm 6000 | . . . 4 | |
17 | 16 | 3ad2ant1 1082 | . . 3 |
18 | 17 | eqeq1d 2624 | . 2 |
19 | 15, 18 | bitrd 268 | 1 supp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 w3a 1037 wceq 1483 wcel 1990 cvv 3200 cun 3572 cin 3573 wss 3574 c0 3915 csn 4177 cxp 5112 cres 5116 wfn 5883 (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: fczsupp0 7324 cantnf0 8572 mdegldg 23826 mdeg0 23830 |
Copyright terms: Public domain | W3C validator |