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Theorem nn0supp 8997
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 8958 . . . 4  |-  NN  =  ( NN0  \  { 0 } )
2 invdif 3288 . . . 4  |-  ( NN0 
i^i  ( _V  \  { 0 } ) )  =  ( NN0  \  { 0 } )
31, 2eqtr4i 2141 . . 3  |-  NN  =  ( NN0  i^i  ( _V 
\  { 0 } ) )
43imaeq2i 4849 . 2  |-  ( `' F " NN )  =  ( `' F " ( NN0  i^i  ( _V  \  { 0 } ) ) )
5 ffun 5245 . . . 4  |-  ( F : I --> NN0  ->  Fun 
F )
6 inpreima 5514 . . . 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 4872 . . . . 5  |-  ( `' F " ( _V 
\  { 0 } ) )  C_  dom  F
9 fdm 5248 . . . . . 6  |-  ( F : I --> NN0  ->  dom 
F  =  I )
10 fimacnv 5517 . . . . . 6  |-  ( F : I --> NN0  ->  ( `' F " NN0 )  =  I )
119, 10eqtr4d 2153 . . . . 5  |-  ( F : I --> NN0  ->  dom 
F  =  ( `' F " NN0 )
)
128, 11sseqtrid 3117 . . . 4  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  C_  ( `' F " NN0 )
)
13 sseqin2 3265 . . . 4  |-  ( ( `' F " ( _V 
\  { 0 } ) )  C_  ( `' F " NN0 )  <->  ( ( `' F " NN0 )  i^i  ( `' F " ( _V 
\  { 0 } ) ) )  =  ( `' F "
( _V  \  {
0 } ) ) )
1412, 13sylib 121 . . 3  |-  ( F : I --> NN0  ->  ( ( `' F " NN0 )  i^i  ( `' F " ( _V 
\  { 0 } ) ) )  =  ( `' F "
( _V  \  {
0 } ) ) )
157, 14eqtrd 2150 . 2  |-  ( F : I --> NN0  ->  ( `' F " ( NN0 
i^i  ( _V  \  { 0 } ) ) )  =  ( `' F " ( _V 
\  { 0 } ) ) )
164, 15syl5req 2163 1  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   _Vcvv 2660    \ cdif 3038    i^i cin 3040    C_ wss 3041   {csn 3497   `'ccnv 4508   dom cdm 4509   "cima 4512   Fun wfun 5087   -->wf 5089   0cc0 7588   NNcn 8688   NN0cn0 8945
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-pre-ltirr 7700  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-inn 8689  df-n0 8946
This theorem is referenced by: (None)
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