Theorem nn0supp 9384

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 9345	. . . 4 |- NN = ( NN0 \ { 0 } )
2		invdif 3424	. . . 4 |- ( NN0 i^i ( _V \ { 0 } ) ) = ( NN0 \ { 0 } )
3	1, 2	eqtr4i 2231	. . 3 |- NN = ( NN0 i^i ( _V \ { 0 } ) )
4	3	imaeq2i 5040	. 2 |- ( `' F " NN ) = ( `' F " ( NN0 i^i ( _V \ { 0 } ) ) )
5		ffun 5449	. . . 4 |- ( F : I --> NN0 -> Fun F )
6		inpreima 5731	. . . 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 5065	. . . . 5 |- ( `' F " ( _V \ { 0 } ) ) C_ dom F
9		fdm 5452	. . . . . 6 |- ( F : I --> NN0 -> dom F = I )
10		fimacnv 5734	. . . . . 6 |- ( F : I --> NN0 -> ( `' F " NN0 ) = I )
11	9, 10	eqtr4d 2243	. . . . 5 |- ( F : I --> NN0 -> dom F = ( `' F " NN0 ) )
12	8, 11	sseqtrid 3252	. . . 4 |- ( F : I --> NN0 -> ( `' F " ( _V \ { 0 } ) ) C_ ( `' F " NN0 ) )
13		sseqin2 3401	. . . 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 2240	. 2 |- ( F : I --> NN0 -> ( `' F " ( NN0 i^i ( _V \ { 0 } ) ) ) = ( `' F " ( _V \ { 0 } ) ) )
16	4, 15	eqtr2id 2253	1 |- ( F : I --> NN0 -> ( `' F " ( _V \ { 0 } ) ) = ( `' F " NN ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2777   \ cdif 3172   i^i cin 3174   C_ wss 3175   {csn 3644   `'ccnv 4693   dom cdm 4694   "cima 4697   Fun wfun 5285   -->wf 5287   0cc0 7962   NNcn 9073   NN0cn0 9332

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1re 8056  ax-addrcl 8059  ax-0lt1 8068  ax-0id 8070  ax-rnegex 8071  ax-pre-ltirr 8074  ax-pre-lttrn 8076  ax-pre-ltadd 8078

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2779  df-sbc 3007  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-fv 5299  df-ov 5972  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-inn 9074  df-n0 9333

This theorem is referenced by: (None)