Theorem fvinim0ffz 10477

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 5479	. . . . . 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 9407	. . . . . . 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 9417	. . . . . . 7 |- ( K e. NN0 -> 0 <_ K )
7	6	adantl 277	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 <_ K )
8		elfz2nn0 10337	. . . . . 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 9401	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
12	11	leidd 8684	. . . . . . 7 |- ( K e. NN0 -> K <_ K )
13		elfz2nn0 10337	. . . . . . 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 5702	. . . . 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 3405	. . 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 3541	. . 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 5779	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` 0 ) e. V )
23	21, 15	ffvelcdmd 5779	. . . 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 3718	. . . 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 3197   (/)c0 3492   {cpr 3668   class class class wbr 4086   "cima 4726   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   0cc0 8022   1c1 8023   <_ cle 8205   NN0cn0 9392   ...cfz 10233  ..^cfzo 10367

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234

This theorem is referenced by: (None)