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Theorem nn0supp 8459
Description: Two ways to write the support of a function on  NN0. (Contributed by Mario Carneiro, 29-Dec-2014.)
Assertion
Ref Expression
nn0supp  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )

Proof of Theorem nn0supp
StepHypRef Expression
1 dfn2 8420 . . . 4  |-  NN  =  ( NN0  \  { 0 } )
2 invdif 3222 . . . 4  |-  ( NN0 
i^i  ( _V  \  { 0 } ) )  =  ( NN0  \  { 0 } )
31, 2eqtr4i 2106 . . 3  |-  NN  =  ( NN0  i^i  ( _V 
\  { 0 } ) )
43imaeq2i 4716 . 2  |-  ( `' F " NN )  =  ( `' F " ( NN0  i^i  ( _V  \  { 0 } ) ) )
5 ffun 5099 . . . 4  |-  ( F : I --> NN0  ->  Fun 
F )
6 inpreima 5345 . . . 4  |-  ( Fun 
F  ->  ( `' F " ( NN0  i^i  ( _V  \  { 0 } ) ) )  =  ( ( `' F " NN0 )  i^i  ( `' F "
( _V  \  {
0 } ) ) ) )
75, 6syl 14 . . 3  |-  ( F : I --> NN0  ->  ( `' F " ( NN0 
i^i  ( _V  \  { 0 } ) ) )  =  ( ( `' F " NN0 )  i^i  ( `' F " ( _V 
\  { 0 } ) ) ) )
8 cnvimass 4738 . . . . 5  |-  ( `' F " ( _V 
\  { 0 } ) )  C_  dom  F
9 fdm 5101 . . . . . 6  |-  ( F : I --> NN0  ->  dom 
F  =  I )
10 fimacnv 5348 . . . . . 6  |-  ( F : I --> NN0  ->  ( `' F " NN0 )  =  I )
119, 10eqtr4d 2118 . . . . 5  |-  ( F : I --> NN0  ->  dom 
F  =  ( `' F " NN0 )
)
128, 11syl5sseq 3056 . . . 4  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  C_  ( `' F " NN0 )
)
13 sseqin2 3201 . . . 4  |-  ( ( `' F " ( _V 
\  { 0 } ) )  C_  ( `' F " NN0 )  <->  ( ( `' F " NN0 )  i^i  ( `' F " ( _V 
\  { 0 } ) ) )  =  ( `' F "
( _V  \  {
0 } ) ) )
1412, 13sylib 120 . . 3  |-  ( F : I --> NN0  ->  ( ( `' F " NN0 )  i^i  ( `' F " ( _V 
\  { 0 } ) ) )  =  ( `' F "
( _V  \  {
0 } ) ) )
157, 14eqtrd 2115 . 2  |-  ( F : I --> NN0  ->  ( `' F " ( NN0 
i^i  ( _V  \  { 0 } ) ) )  =  ( `' F " ( _V 
\  { 0 } ) ) )
164, 15syl5req 2128 1  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   _Vcvv 2610    \ cdif 2979    i^i cin 2981    C_ wss 2982   {csn 3416   `'ccnv 4390   dom cdm 4391   "cima 4394   Fun wfun 4946   -->wf 4948   0cc0 7095   NNcn 8158   NN0cn0 8407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1re 7184  ax-addrcl 7187  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-pre-ltirr 7202  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-ov 5566  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-inn 8159  df-n0 8408
This theorem is referenced by: (None)
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