Theorem fvinim0ffz 10677

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 5536	. . . . . 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 9589	. . . . . . 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 9599	. . . . . . 7 |- ( K e. NN0 -> 0 <_ K )
7	6	adantl 277	. . . . . 6 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 <_ K )
8		elfz2nn0 10536	. . . . . 6 |- ( 0 e. ( 0 ... K ) <-> ( 0 e. NN0 /\ K e. NN0 /\ 0 <_ K ) )
9	4, 5, 7, 8	syl3anbrc 1212	. . . . 5 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> 0 e. ( 0 ... K ) )
10		id 19	. . . . . . 7 |- ( K e. NN0 -> K e. NN0 )
11		nn0re 9583	. . . . . . . 8 |- ( K e. NN0 -> K e. RR )
12	11	leidd 8849	. . . . . . 7 |- ( K e. NN0 -> K <_ K )
13		elfz2nn0 10536	. . . . . . 7 |- ( K e. ( 0 ... K ) <-> ( K e. NN0 /\ K e. NN0 /\ K <_ K ) )
14	10, 10, 12, 13	syl3anbrc 1212	. . . . . 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 5766	. . . . 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 1278	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F " { 0 , K } ) = { ( F ` 0 ) , ( F ` K ) } )
18	17	ineq1d 3431	. . 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 2247	. 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 3573	. . 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 5847	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` 0 ) e. V )
23	21, 15	ffvelcdmd 5847	. . . 4 |- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( F ` K ) e. V )
24		eleq1 2301	. . . . . . 7 |- ( v = ( F ` 0 ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
25	24	notbid 677	. . . . . 6 |- ( v = ( F ` 0 ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` 0 ) e. ( F " ( 1..^ K ) ) ) )
26		df-nel 2516	. . . . . 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 2301	. . . . . . 7 |- ( v = ( F ` K ) -> ( v e. ( F " ( 1..^ K ) ) <-> ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
29	28	notbid 677	. . . . . 6 |- ( v = ( F ` K ) -> ( -. v e. ( F " ( 1..^ K ) ) <-> -. ( F ` K ) e. ( F " ( 1..^ K ) ) ) )
30		df-nel 2516	. . . . . 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 3760	. . . 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 415	. . 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 ) ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   e/ wnel 2515   A.wral 2528   i^i cin 3219   (/)c0 3520   {cpr 3710   class class class wbr 4131   "cima 4779   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   0cc0 8185   1c1 8186   <_ cle 8367   NN0cn0 9574   ...cfz 10428  ..^cfzo 10566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429

This theorem is used by: (None)