Theorem resunimafz0 11223

Description: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

Hypotheses

Ref	Expression
resunimafz0.i	|- ( ph -> Fun I )
resunimafz0.f	|- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
resunimafz0.n	|- ( ph -> N e. ( 0..^ (♯ ` F ) ) )

Assertion

Ref	Expression
resunimafz0	|- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )

Proof of Theorem resunimafz0

Step	Hyp	Ref	Expression
1		imaundi 5180	. . . . 5 |- ( F " ( ( 0..^ N ) u. { N } ) ) = ( ( F " ( 0..^ N ) ) u. ( F " { N } ) )
2		resunimafz0.n	. . . . . . . . 9 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
3		elfzonn0 10547	. . . . . . . . 9 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. NN0 )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. NN0 )
5		elnn0uz 9910	. . . . . . . 8 |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
6	4, 5	sylib 122	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` 0 ) )
7		fzisfzounsn 10604	. . . . . . 7 |- ( N e. ( ZZ>= ` 0 ) -> ( 0 ... N ) = ( ( 0..^ N ) u. { N } ) )
8	6, 7	syl 14	. . . . . 6 |- ( ph -> ( 0 ... N ) = ( ( 0..^ N ) u. { N } ) )
9	8	imaeq2d 5106	. . . . 5 |- ( ph -> ( F " ( 0 ... N ) ) = ( F " ( ( 0..^ N ) u. { N } ) ) )
10		resunimafz0.f	. . . . . . . 8 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
11	10	ffnd 5514	. . . . . . 7 |- ( ph -> F Fn ( 0..^ (♯ ` F ) ) )
12		fnsnfv 5741	. . . . . . 7 |- ( ( F Fn ( 0..^ (♯ ` F ) ) /\ N e. ( 0..^ (♯ ` F ) ) ) -> { ( F ` N ) } = ( F " { N } ) )
13	11, 2, 12	syl2anc 411	. . . . . 6 |- ( ph -> { ( F ` N ) } = ( F " { N } ) )
14	13	uneq2d 3377	. . . . 5 |- ( ph -> ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) = ( ( F " ( 0..^ N ) ) u. ( F " { N } ) ) )
15	1, 9, 14	3eqtr4a 2293	. . . 4 |- ( ph -> ( F " ( 0 ... N ) ) = ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) )
16	15	reseq2d 5043	. . 3 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( I |` ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) ) )
17		resundi 5056	. . 3 |- ( I |` ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) )
18	16, 17	eqtrdi 2283	. 2 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) ) )
19		resunimafz0.i	. . . . 5 |- ( ph -> Fun I )
20		funfn 5387	. . . . 5 |- ( Fun I <-> I Fn dom I )
21	19, 20	sylib 122	. . . 4 |- ( ph -> I Fn dom I )
22	10, 2	ffvelcdmd 5818	. . . 4 |- ( ph -> ( F ` N ) e. dom I )
23		fnressn 5875	. . . 4 |- ( ( I Fn dom I /\ ( F ` N ) e. dom I ) -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
24	21, 22, 23	syl2anc 411	. . 3 |- ( ph -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
25	24	uneq2d 3377	. 2 |- ( ph -> ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
26	18, 25	eqtrd 2267	1 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   u. cun 3212   {csn 3694   <.cop 3697   dom cdm 4754   |` cres 4756   "cima 4757   Fun wfun 5351   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   0cc0 8143   NN0cn0 9513   ZZ>=cuz 9871   ...cfz 10361  ..^cfzo 10498  ♯chash 11163

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499

This theorem is referenced by:  trlsegvdegfi  16588