Theorem fvinim0ffz 10533

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 5489	. . . . . 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 9459	. . . . . . 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 9469	. . . . . . 7 |- ( K e. NN0 -> 0 <_ K )
7	6	adantl 277	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 <_ K )
8		elfz2nn0 10392	. . . . . 6 |- ( 0 e. ( 0 ... K ) <-> ( 0 e. NN0 /\ K e. NN0 /\ 0 <_ K ) )
9	4, 5, 7, 8	syl3anbrc 1208	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 e. ( 0 ... K ) )
10		id 19	. . . . . . 7 |- ( K e. NN0 -> K e. NN0 )
11		nn0re 9453	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
12	11	leidd 8736	. . . . . . 7 |- ( K e. NN0 -> K <_ K )
13		elfz2nn0 10392	. . . . . . 7 |- ( K e. ( 0 ... K ) <-> ( K e. NN0 /\ K e. NN0 /\ K <_ K ) )
14	10, 10, 12, 13	syl3anbrc 1208	. . . . . 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 5715	. . . . 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 1274	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F " { 0 , K } ) = { ( F ` 0 ) , ( F ` K ) } )
18	17	ineq1d 3409	. . 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 2240	. 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 3545	. . 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 5791	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` 0 ) e. V )
23	21, 15	ffvelcdmd 5791	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` K ) e. V )
24		eleq1 2294	. . . . . . 7 |- ( v = ( F ` 0 ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
25	24	notbid 673	. . . . . 6 |- ( v = ( F ` 0 ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
26		df-nel 2499	. . . . . 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 2294	. . . . . . 7 |- ( v = ( F ` K ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
29	28	notbid 673	. . . . . 6 |- ( v = ( F ` K ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
30		df-nel 2499	. . . . . 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 3724	. . . 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 1398   e. wcel 2202   e/ wnel 2498   A.wral 2511   i^i cin 3200   (/)c0 3496   {cpr 3674   class class class wbr 4093   "cima 4734   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   0cc0 8075   1c1 8076   <_ cle 8257   NN0cn0 9444   ...cfz 10288  ..^cfzo 10422

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289

This theorem is referenced by: (None)