Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fvinim0ffz | Unicode version |
Description: The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.) |
Ref | Expression |
---|---|
fvinim0ffz | ..^ ..^ ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5319 | . . . . . 6 | |
2 | 1 | adantr 274 | . . . . 5 |
3 | 0nn0 9105 | . . . . . . 7 | |
4 | 3 | a1i 9 | . . . . . 6 |
5 | simpr 109 | . . . . . 6 | |
6 | nn0ge0 9115 | . . . . . . 7 | |
7 | 6 | adantl 275 | . . . . . 6 |
8 | elfz2nn0 10014 | . . . . . 6 | |
9 | 4, 5, 7, 8 | syl3anbrc 1166 | . . . . 5 |
10 | id 19 | . . . . . . 7 | |
11 | nn0re 9099 | . . . . . . . 8 | |
12 | 11 | leidd 8389 | . . . . . . 7 |
13 | elfz2nn0 10014 | . . . . . . 7 | |
14 | 10, 10, 12, 13 | syl3anbrc 1166 | . . . . . 6 |
15 | 14 | adantl 275 | . . . . 5 |
16 | fnimapr 5528 | . . . . 5 | |
17 | 2, 9, 15, 16 | syl3anc 1220 | . . . 4 |
18 | 17 | ineq1d 3307 | . . 3 ..^ ..^ |
19 | 18 | eqeq1d 2166 | . 2 ..^ ..^ |
20 | disj 3442 | . . 3 ..^ ..^ | |
21 | simpl 108 | . . . . 5 | |
22 | 21, 9 | ffvelrnd 5603 | . . . 4 |
23 | 21, 15 | ffvelrnd 5603 | . . . 4 |
24 | eleq1 2220 | . . . . . . 7 ..^ ..^ | |
25 | 24 | notbid 657 | . . . . . 6 ..^ ..^ |
26 | df-nel 2423 | . . . . . 6 ..^ ..^ | |
27 | 25, 26 | bitr4di 197 | . . . . 5 ..^ ..^ |
28 | eleq1 2220 | . . . . . . 7 ..^ ..^ | |
29 | 28 | notbid 657 | . . . . . 6 ..^ ..^ |
30 | df-nel 2423 | . . . . . 6 ..^ ..^ | |
31 | 29, 30 | bitr4di 197 | . . . . 5 ..^ ..^ |
32 | 27, 31 | ralprg 3610 | . . . 4 ..^ ..^ ..^ |
33 | 22, 23, 32 | syl2anc 409 | . . 3 ..^ ..^ ..^ |
34 | 20, 33 | syl5bb 191 | . 2 ..^ ..^ ..^ |
35 | 19, 34 | bitrd 187 | 1 ..^ ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wnel 2422 wral 2435 cin 3101 c0 3394 cpr 3561 class class class wbr 3965 cima 4589 wfn 5165 wf 5166 cfv 5170 (class class class)co 5824 cc0 7732 c1 7733 cle 7913 cn0 9090 cfz 9912 ..^cfzo 10041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-uz 9440 df-fz 9913 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |