Theorem resunimafz0 10976

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 5095	. . . . 5 |- ( F " ( ( 0..^ N ) u. { N } ) ) = ( ( F " ( 0..^ N ) ) u. ( F " { N } ) )
2		resunimafz0.n	. . . . . . . . 9 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
3		elfzonn0 10310	. . . . . . . . 9 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. NN0 )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. NN0 )
5		elnn0uz 9686	. . . . . . . 8 |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
6	4, 5	sylib 122	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` 0 ) )
7		fzisfzounsn 10365	. . . . . . 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 5022	. . . . 5 |- ( ph -> ( F " ( 0 ... N ) ) = ( F " ( ( 0..^ N ) u. { N } ) ) )
10		resunimafz0.f	. . . . . . . 8 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
11	10	ffnd 5426	. . . . . . 7 |- ( ph -> F Fn ( 0..^ (♯ ` F ) ) )
12		fnsnfv 5638	. . . . . . 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 3327	. . . . 5 |- ( ph -> ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) = ( ( F " ( 0..^ N ) ) u. ( F " { N } ) ) )
15	1, 9, 14	3eqtr4a 2264	. . . 4 |- ( ph -> ( F " ( 0 ... N ) ) = ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) )
16	15	reseq2d 4959	. . 3 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( I |` ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) ) )
17		resundi 4972	. . 3 |- ( I |` ( ( F " ( 0..^ N ) ) u. { ( F ` N ) } ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) )
18	16, 17	eqtrdi 2254	. 2 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) ) )
19		resunimafz0.i	. . . . 5 |- ( ph -> Fun I )
20		funfn 5301	. . . . 5 |- ( Fun I <-> I Fn dom I )
21	19, 20	sylib 122	. . . 4 |- ( ph -> I Fn dom I )
22	10, 2	ffvelcdmd 5716	. . . 4 |- ( ph -> ( F ` N ) e. dom I )
23		fnressn 5770	. . . 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 3327	. 2 |- ( ph -> ( ( I |` ( F " ( 0..^ N ) ) ) u. ( I |` { ( F ` N ) } ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
26	18, 25	eqtrd 2238	1 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   u. cun 3164   {csn 3633   <.cop 3636   dom cdm 4675   |` cres 4677   "cima 4678   Fun wfun 5265   Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5944   0cc0 7925   NN0cn0 9295   ZZ>=cuz 9648   ...cfz 10130  ..^cfzo 10264  ♯chash 10920

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-fzo 10265

This theorem is referenced by: (None)