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Theorem resunimafz0 10792
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
StepHypRef Expression
1 imaundi 5037 . . . . 5  |-  ( F
" ( ( 0..^ N )  u.  { N } ) )  =  ( ( F "
( 0..^ N ) )  u.  ( F
" { N }
) )
2 resunimafz0.n . . . . . . . . 9  |-  ( ph  ->  N  e.  ( 0..^ ( `  F )
) )
3 elfzonn0 10169 . . . . . . . . 9  |-  ( N  e.  ( 0..^ ( `  F ) )  ->  N  e.  NN0 )
42, 3syl 14 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
5 elnn0uz 9551 . . . . . . . 8  |-  ( N  e.  NN0  <->  N  e.  ( ZZ>=
`  0 ) )
64, 5sylib 122 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
7 fzisfzounsn 10219 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( 0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
86, 7syl 14 . . . . . 6  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0..^ N )  u. 
{ N } ) )
98imaeq2d 4966 . . . . 5  |-  ( ph  ->  ( F " (
0 ... N ) )  =  ( F "
( ( 0..^ N )  u.  { N } ) ) )
10 resunimafz0.f . . . . . . . 8  |-  ( ph  ->  F : ( 0..^ ( `  F )
) --> dom  I )
1110ffnd 5362 . . . . . . 7  |-  ( ph  ->  F  Fn  ( 0..^ ( `  F )
) )
12 fnsnfv 5571 . . . . . . 7  |-  ( ( F  Fn  ( 0..^ ( `  F )
)  /\  N  e.  ( 0..^ ( `  F
) ) )  ->  { ( F `  N ) }  =  ( F " { N } ) )
1311, 2, 12syl2anc 411 . . . . . 6  |-  ( ph  ->  { ( F `  N ) }  =  ( F " { N } ) )
1413uneq2d 3289 . . . . 5  |-  ( ph  ->  ( ( F "
( 0..^ N ) )  u.  { ( F `  N ) } )  =  ( ( F " (
0..^ N ) )  u.  ( F " { N } ) ) )
151, 9, 143eqtr4a 2236 . . . 4  |-  ( ph  ->  ( F " (
0 ... N ) )  =  ( ( F
" ( 0..^ N ) )  u.  {
( F `  N
) } ) )
1615reseq2d 4903 . . 3  |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( I  |`  ( ( F "
( 0..^ N ) )  u.  { ( F `  N ) } ) ) )
17 resundi 4916 . . 3  |-  ( I  |`  ( ( F "
( 0..^ N ) )  u.  { ( F `  N ) } ) )  =  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  ( I  |`  { ( F `  N ) } ) )
1816, 17eqtrdi 2226 . 2  |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( ( I  |`  ( F " (
0..^ N ) ) )  u.  ( I  |`  { ( F `  N ) } ) ) )
19 resunimafz0.i . . . . 5  |-  ( ph  ->  Fun  I )
20 funfn 5242 . . . . 5  |-  ( Fun  I  <->  I  Fn  dom  I )
2119, 20sylib 122 . . . 4  |-  ( ph  ->  I  Fn  dom  I
)
2210, 2ffvelcdmd 5648 . . . 4  |-  ( ph  ->  ( F `  N
)  e.  dom  I
)
23 fnressn 5698 . . . 4  |-  ( ( I  Fn  dom  I  /\  ( F `  N
)  e.  dom  I
)  ->  ( I  |` 
{ ( F `  N ) } )  =  { <. ( F `  N ) ,  ( I `  ( F `  N ) ) >. } )
2421, 22, 23syl2anc 411 . . 3  |-  ( ph  ->  ( I  |`  { ( F `  N ) } )  =  { <. ( F `  N
) ,  ( I `
 ( F `  N ) ) >. } )
2524uneq2d 3289 . 2  |-  ( ph  ->  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  ( I  |`  { ( F `  N ) } ) )  =  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  { <. ( F `  N
) ,  ( I `
 ( F `  N ) ) >. } ) )
2618, 25eqtrd 2210 1  |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( ( I  |`  ( F " (
0..^ N ) ) )  u.  { <. ( F `  N ) ,  ( I `  ( F `  N ) ) >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    u. cun 3127   {csn 3591   <.cop 3594   dom cdm 4623    |` cres 4625   "cima 4626   Fun wfun 5206    Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869   0cc0 7799   NN0cn0 9162   ZZ>=cuz 9514   ...cfz 9992  ..^cfzo 10125  ♯chash 10736
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-distr 7903  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-cnre 7910  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913  ax-pre-apti 7914  ax-pre-ltadd 7915
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985  df-sub 8117  df-neg 8118  df-inn 8906  df-n0 9163  df-z 9240  df-uz 9515  df-fz 9993  df-fzo 10126
This theorem is referenced by: (None)
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