Theorem fvinim0ffz 10459

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.)

Assertion

Ref	Expression
fvinim0ffz	|- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( ( F " { 0 , K } ) i^i ( F " ( 1..^ K ) ) ) = (/) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )

Proof of Theorem fvinim0ffz

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5473	. . . . . 6 |- ( F : ( 0 ... K ) --> V -> F Fn ( 0 ... K ) )
2	1	adantr 276	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> F Fn ( 0 ... K ) )
3		0nn0 9395	. . . . . . 7 |- 0 e. NN0
4	3	a1i 9	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 e. NN0 )
5		simpr 110	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> K e. NN0 )
6		nn0ge0 9405	. . . . . . 7 |- ( K e. NN0 -> 0 <_ K )
7	6	adantl 277	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 <_ K )
8		elfz2nn0 10320	. . . . . 6 |- ( 0 e. ( 0 ... K ) <-> ( 0 e. NN0 /\ K e. NN0 /\ 0 <_ K ) )
9	4, 5, 7, 8	syl3anbrc 1205	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 e. ( 0 ... K ) )
10		id 19	. . . . . . 7 |- ( K e. NN0 -> K e. NN0 )
11		nn0re 9389	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
12	11	leidd 8672	. . . . . . 7 |- ( K e. NN0 -> K <_ K )
13		elfz2nn0 10320	. . . . . . 7 |- ( K e. ( 0 ... K ) <-> ( K e. NN0 /\ K e. NN0 /\ K <_ K ) )
14	10, 10, 12, 13	syl3anbrc 1205	. . . . . 6 |- ( K e. NN0 -> K e. ( 0 ... K ) )
15	14	adantl 277	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> K e. ( 0 ... K ) )
16		fnimapr 5696	. . . . 5 |- ( ( F Fn ( 0 ... K ) /\ 0 e. ( 0 ... K ) /\ K e. ( 0 ... K ) ) -> ( F " { 0 , K } ) = { ( F ` 0 ) , ( F ` K ) } )
17	2, 9, 15, 16	syl3anc 1271	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F " { 0 , K } ) = { ( F ` 0 ) , ( F ` K ) } )
18	17	ineq1d 3404	. . 3 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( F " { 0 , K } ) i^i ( F " ( 1..^ K ) ) ) = ( { ( F ` 0 ) , ( F ` K ) } i^i ( F " ( 1..^ K ) ) ) )
19	18	eqeq1d 2238	. 2 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( ( F " { 0 , K } ) i^i ( F " ( 1..^ K ) ) ) = (/) <-> ( { ( F ` 0 ) , ( F ` K ) } i^i ( F " ( 1..^ K ) ) ) = (/) ) )
20		disj 3540	. . 3 |- ( ( { ( F ` 0 ) , ( F ` K ) } i^i ( F " ( 1..^ K ) ) ) = (/) <-> A. v e. { ( F ` 0 ) , ( F ` K ) } -. v e. ( F " ( 1..^ K ) ) )
21		simpl 109	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> F : ( 0 ... K ) --> V )
22	21, 9	ffvelcdmd 5773	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` 0 ) e. V )
23	21, 15	ffvelcdmd 5773	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` K ) e. V )
24		eleq1 2292	. . . . . . 7 |- ( v = ( F ` 0 ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
25	24	notbid 671	. . . . . 6 |- ( v = ( F ` 0 ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
26		df-nel 2496	. . . . . 6 |- ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) <-> -. ( F ` 0 ) e. ( F " ( 1..^ K ) ) )
27	25, 26	bitr4di 198	. . . . 5 |- ( v = ( F ` 0 ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> ( F ` 0 ) e/ ( F " ( 1..^ K ) ) ) )
28		eleq1 2292	. . . . . . 7 |- ( v = ( F ` K ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
29	28	notbid 671	. . . . . 6 |- ( v = ( F ` K ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
30		df-nel 2496	. . . . . 6 |- ( ( F ` K ) e/ ( F " ( 1..^ K ) ) <-> -. ( F ` K ) e. ( F " ( 1..^ K ) ) )
31	29, 30	bitr4di 198	. . . . 5 |- ( v = ( F ` K ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> ( F ` K ) e/ ( F " ( 1..^ K ) ) ) )
32	27, 31	ralprg 3717	. . . 4 |- ( ( ( F ` 0 ) e. V /\ ( F ` K ) e. V ) -> ( A. v e. { ( F ` 0 ) , ( F ` K ) } -. v e. ( F " ( 1..^ K ) ) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )
33	22, 23, 32	syl2anc 411	. . 3 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( A. v e. { ( F ` 0 ) , ( F ` K ) } -. v e. ( F " ( 1..^ K ) ) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )
34	20, 33	bitrid 192	. 2 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( { ( F ` 0 ) , ( F ` K ) } i^i ( F " ( 1..^ K ) ) ) = (/) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )
35	19, 34	bitrd 188	1 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( ( F " { 0 , K } ) i^i ( F " ( 1..^ K ) ) ) = (/) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   e/ wnel 2495   A.wral 2508   i^i cin 3196   (/)c0 3491   {cpr 3667   class class class wbr 4083   "cima 4722   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6007   0cc0 8010   1c1 8011   <_ cle 8193   NN0cn0 9380   ...cfz 10216  ..^cfzo 10350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217

This theorem is referenced by: (None)