Theorem resunimafz0 11274

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 5200	. . . . 5 |- ( F " ( ( 0..^ N ) u. { N } ) ) = ( ( F " ( 0..^ N ) ) u. ( F " { N } ) )
2		resunimafz0.n	. . . . . . . . 9 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
3		elfzonn0 10598	. . . . . . . . 9 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. NN0 )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. NN0 )
5		elnn0uz 9960	. . . . . . . 8 |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
6	4, 5	sylib 122	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` 0 ) )
7		fzisfzounsn 10655	. . . . . . 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 5126	. . . . 5 |- ( ph -> ( F " ( 0 ... N ) ) = ( F " ( ( 0..^ N ) u. { N } ) ) )
10		resunimafz0.f	. . . . . . . 8 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
11	10	ffnd 5534	. . . . . . 7 |- ( ph -> F Fn ( 0..^ (♯ ` F ) ) )
12		fnsnfv 5762	. . . . . . 7 |- ( ( F Fn ( 0..^ (♯ ` F ) ) /\ N e. ( 0..^ (♯ ` F ) ) ) -> { ( F ` N ) } = ( F " { N } ) )
13	11, 2, 12	syl2anc 415	. . . . . 6 |- ( ph -> { ( F ` N ) } = ( F " { N } ) )
14	13	uneq2d 3383	. . . . 5 |- ( ph -> ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) = ( ( F " ( 0..^ N ) ) u. ( F " { N } ) ) )
15	1, 9, 14	3eqtr4a 2297	. . . 4 |- ( ph -> ( F " ( 0 ... N ) ) = ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) )
16	15	reseq2d 5063	. . 3 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( I |` ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) ) )
17		resundi 5076	. . 3 |- ( I |` ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) )
18	16, 17	eqtrdi 2287	. 2 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) ) )
19		resunimafz0.i	. . . . 5 |- ( ph -> Fun I )
20		funfn 5407	. . . . 5 |- ( Fun I <-> I Fn dom I )
21	19, 20	sylib 122	. . . 4 |- ( ph -> I Fn dom I )
22	10, 2	ffvelcdmd 5844	. . . 4 |- ( ph -> ( F ` N ) e. dom I )
23		fnressn 5901	. . . 4 |- ( ( I Fn dom I /\ ( F ` N ) e. dom I ) -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
24	21, 22, 23	syl2anc 415	. . 3 |- ( ph -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
25	24	uneq2d 3383	. 2 |- ( ph -> ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
26	18, 25	eqtrd 2271	1 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   u. cun 3218   {csn 3709   <.cop 3712   dom cdm 4774   |` cres 4776   "cima 4777   Fun wfun 5371   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   0cc0 8179   NN0cn0 9563   ZZ>=cuz 9921   ...cfz 10411  ..^cfzo 10549  ♯chash 11214

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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

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-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550

This theorem is used by:  trlsegvdegfi  16708