Theorem fvinim0ffz 10238

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 5365	. . . . . 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 9189	. . . . . . 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 9199	. . . . . . 7 |- ( K e. NN0 -> 0 <_ K )
7	6	adantl 277	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 <_ K )
8		elfz2nn0 10109	. . . . . 6 |- ( 0 e. ( 0 ... K ) <-> ( 0 e. NN0 /\ K e. NN0 /\ 0 <_ K ) )
9	4, 5, 7, 8	syl3anbrc 1181	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 e. ( 0 ... K ) )
10		id 19	. . . . . . 7 |- ( K e. NN0 -> K e. NN0 )
11		nn0re 9183	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
12	11	leidd 8469	. . . . . . 7 |- ( K e. NN0 -> K <_ K )
13		elfz2nn0 10109	. . . . . . 7 |- ( K e. ( 0 ... K ) <-> ( K e. NN0 /\ K e. NN0 /\ K <_ K ) )
14	10, 10, 12, 13	syl3anbrc 1181	. . . . . 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 5576	. . . . 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 1238	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F " { 0 , K } ) = { ( F ` 0 ) , ( F ` K ) } )
18	17	ineq1d 3335	. . 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 2186	. 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 3471	. . 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 5652	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` 0 ) e. V )
23	21, 15	ffvelcdmd 5652	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` K ) e. V )
24		eleq1 2240	. . . . . . 7 |- ( v = ( F ` 0 ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
25	24	notbid 667	. . . . . 6 |- ( v = ( F ` 0 ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
26		df-nel 2443	. . . . . 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 2240	. . . . . . 7 |- ( v = ( F ` K ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
29	28	notbid 667	. . . . . 6 |- ( v = ( F ` K ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
30		df-nel 2443	. . . . . 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 3643	. . . 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 1353   e. wcel 2148   e/ wnel 2442   A.wral 2455   i^i cin 3128   (/)c0 3422   {cpr 3593   class class class wbr 4003   "cima 4629   Fn wfn 5211   -->wf 5212   ` cfv 5216  (class class class)co 5874   0cc0 7810   1c1 7811   <_ cle 7991   NN0cn0 9174   ...cfz 10006  ..^cfzo 10139

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-fz 10007

This theorem is referenced by: (None)