Theorem nn0supp 9421

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

Step	Hyp	Ref	Expression
1		dfn2 9382	. . . 4 |- NN = ( NN0 \ { 0 } )
2		invdif 3446	. . . 4 |- ( NN0 i^i ( _V \ { 0 } ) ) = ( NN0 \ { 0 } )
3	1, 2	eqtr4i 2253	. . 3 |- NN = ( NN0 i^i ( _V \ { 0 } ) )
4	3	imaeq2i 5066	. 2 |- ( `' F " NN ) = ( `' F " ( NN0 i^i ( _V \ { 0 } ) ) )
5		ffun 5476	. . . 4 |- ( F : I --> NN0 -> Fun F )
6		inpreima 5761	. . . 4 |- ( Fun F -> ( `' F " ( NN0 i^i ( _V \ { 0 } ) ) ) = ( ( `' F " NN0 ) i^i ( `' F " ( _V \ { 0 } ) ) ) )
7	5, 6	syl 14	. . 3 |- ( F : I --> NN0 -> ( `' F " ( NN0 i^i ( _V \ { 0 } ) ) ) = ( ( `' F " NN0 ) i^i ( `' F " ( _V \ { 0 } ) ) ) )
8		cnvimass 5091	. . . . 5 |- ( `' F " ( _V \ { 0 } ) ) C_ dom F
9		fdm 5479	. . . . . 6 |- ( F : I --> NN0 -> dom F = I )
10		fimacnv 5764	. . . . . 6 |- ( F : I --> NN0 -> ( `' F " NN0 ) = I )
11	9, 10	eqtr4d 2265	. . . . 5 |- ( F : I --> NN0 -> dom F = ( `' F " NN0 ) )
12	8, 11	sseqtrid 3274	. . . 4 |- ( F : I --> NN0 -> ( `' F " ( _V \ { 0 } ) ) C_ ( `' F " NN0 ) )
13		sseqin2 3423	. . . 4 |- ( ( `' F " ( _V \ { 0 } ) ) C_ ( `' F " NN0 ) <-> ( ( `' F " NN0 ) i^i ( `' F " ( _V \ { 0 } ) ) ) = ( `' F " ( _V \ { 0 } ) ) )
14	12, 13	sylib 122	. . 3 |- ( F : I --> NN0 -> ( ( `' F " NN0 ) i^i ( `' F " ( _V \ { 0 } ) ) ) = ( `' F " ( _V \ { 0 } ) ) )
15	7, 14	eqtrd 2262	. 2 |- ( F : I --> NN0 -> ( `' F " ( NN0 i^i ( _V \ { 0 } ) ) ) = ( `' F " ( _V \ { 0 } ) ) )
16	4, 15	eqtr2id 2275	1 |- ( F : I --> NN0 -> ( `' F " ( _V \ { 0 } ) ) = ( `' F " NN ) )

Syntax hints:   -> wi 4   = wceq 1395   _Vcvv 2799   \ cdif 3194   i^i cin 3196   C_ wss 3197   {csn 3666   `'ccnv 4718   dom cdm 4719   "cima 4722   Fun wfun 5312   -->wf 5314   0cc0 7999   NNcn 9110   NN0cn0 9369

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-inn 9111  df-n0 9370

This theorem is referenced by: (None)