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Theorem resunimafz0 10581
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 4951 . . . . 5  |-  ( F
" ( ( 0..^ N )  u.  { N } ) )  =  ( ( F "
( 0..^ N ) )  u.  ( F
" { N }
) )
2 resunimafz0.n . . . . . . . . 9  |-  ( ph  ->  N  e.  ( 0..^ ( `  F )
) )
3 elfzonn0 9970 . . . . . . . . 9  |-  ( N  e.  ( 0..^ ( `  F ) )  ->  N  e.  NN0 )
42, 3syl 14 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
5 elnn0uz 9370 . . . . . . . 8  |-  ( N  e.  NN0  <->  N  e.  ( ZZ>=
`  0 ) )
64, 5sylib 121 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
7 fzisfzounsn 10020 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( 0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
86, 7syl 14 . . . . . 6  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0..^ N )  u. 
{ N } ) )
98imaeq2d 4881 . . . . 5  |-  ( ph  ->  ( F " (
0 ... N ) )  =  ( F "
( ( 0..^ N )  u.  { N } ) ) )
10 resunimafz0.f . . . . . . . 8  |-  ( ph  ->  F : ( 0..^ ( `  F )
) --> dom  I )
1110ffnd 5273 . . . . . . 7  |-  ( ph  ->  F  Fn  ( 0..^ ( `  F )
) )
12 fnsnfv 5480 . . . . . . 7  |-  ( ( F  Fn  ( 0..^ ( `  F )
)  /\  N  e.  ( 0..^ ( `  F
) ) )  ->  { ( F `  N ) }  =  ( F " { N } ) )
1311, 2, 12syl2anc 408 . . . . . 6  |-  ( ph  ->  { ( F `  N ) }  =  ( F " { N } ) )
1413uneq2d 3230 . . . . 5  |-  ( ph  ->  ( ( F "
( 0..^ N ) )  u.  { ( F `  N ) } )  =  ( ( F " (
0..^ N ) )  u.  ( F " { N } ) ) )
151, 9, 143eqtr4a 2198 . . . 4  |-  ( ph  ->  ( F " (
0 ... N ) )  =  ( ( F
" ( 0..^ N ) )  u.  {
( F `  N
) } ) )
1615reseq2d 4819 . . 3  |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( I  |`  ( ( F "
( 0..^ N ) )  u.  { ( F `  N ) } ) ) )
17 resundi 4832 . . 3  |-  ( I  |`  ( ( F "
( 0..^ N ) )  u.  { ( F `  N ) } ) )  =  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  ( I  |`  { ( F `  N ) } ) )
1816, 17syl6eq 2188 . 2  |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( ( I  |`  ( F " (
0..^ N ) ) )  u.  ( I  |`  { ( F `  N ) } ) ) )
19 resunimafz0.i . . . . 5  |-  ( ph  ->  Fun  I )
20 funfn 5153 . . . . 5  |-  ( Fun  I  <->  I  Fn  dom  I )
2119, 20sylib 121 . . . 4  |-  ( ph  ->  I  Fn  dom  I
)
2210, 2ffvelrnd 5556 . . . 4  |-  ( ph  ->  ( F `  N
)  e.  dom  I
)
23 fnressn 5606 . . . 4  |-  ( ( I  Fn  dom  I  /\  ( F `  N
)  e.  dom  I
)  ->  ( I  |` 
{ ( F `  N ) } )  =  { <. ( F `  N ) ,  ( I `  ( F `  N ) ) >. } )
2421, 22, 23syl2anc 408 . . 3  |-  ( ph  ->  ( I  |`  { ( F `  N ) } )  =  { <. ( F `  N
) ,  ( I `
 ( F `  N ) ) >. } )
2524uneq2d 3230 . 2  |-  ( ph  ->  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  ( I  |`  { ( F `  N ) } ) )  =  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  { <. ( F `  N
) ,  ( I `
 ( F `  N ) ) >. } ) )
2618, 25eqtrd 2172 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 1331    e. wcel 1480    u. cun 3069   {csn 3527   <.cop 3530   dom cdm 4539    |` cres 4541   "cima 4542   Fun wfun 5117    Fn wfn 5118   -->wf 5119   ` cfv 5123  (class class class)co 5774   0cc0 7627   NN0cn0 8984   ZZ>=cuz 9333   ...cfz 9797  ..^cfzo 9926  ♯chash 10528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-fzo 9927
This theorem is referenced by: (None)
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