Theorem frnsuppeq 7307

Description: Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)

Assertion

Ref	Expression
frnsuppeq	|- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )

Proof of Theorem frnsuppeq

Step	Hyp	Ref	Expression
1		fex 6490	. . . . . . 7 |- ( ( F : I --> S /\ I e. V ) -> F e. _V )
2	1	expcom 451	. . . . . 6 |- ( I e. V -> ( F : I --> S -> F e. _V ) )
3	2	adantr 481	. . . . 5 |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> F e. _V ) )
4	3	imp 445	. . . 4 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> F e. _V )
5		simplr 792	. . . 4 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> Z e. W )
6		suppimacnv 7306	. . . 4 |- ( ( F e. _V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
7	4, 5, 6	syl2anc 693	. . 3 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
8		invdif 3868	. . . . . 6 |- ( S i^i ( _V \ { Z } ) ) = ( S \ { Z } )
9	8	imaeq2i 5464	. . . . 5 |- ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( S \ { Z } ) )
10		ffun 6048	. . . . . . 7 |- ( F : I --> S -> Fun F )
11		inpreima 6342	. . . . . . 7 |- ( Fun F -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) )
12	10, 11	syl 17	. . . . . 6 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) )
13		cnvimass 5485	. . . . . . . 8 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
14		fdm 6051	. . . . . . . . 9 |- ( F : I --> S -> dom F = I )
15		fimacnv 6347	. . . . . . . . 9 |- ( F : I --> S -> ( `' F " S ) = I )
16	14, 15	eqtr4d 2659	. . . . . . . 8 |- ( F : I --> S -> dom F = ( `' F " S ) )
17	13, 16	syl5sseq 3653	. . . . . . 7 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) )
18		sseqin2 3817	. . . . . . 7 |- ( ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) <-> ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
19	17, 18	sylib 208	. . . . . 6 |- ( F : I --> S -> ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
20	12, 19	eqtrd 2656	. . . . 5 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
21	9, 20	syl5reqr 2671	. . . 4 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) = ( `' F " ( S \ { Z } ) ) )
22	21	adantl 482	. . 3 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> ( `' F " ( _V \ { Z } ) ) = ( `' F " ( S \ { Z } ) ) )
23	7, 22	eqtrd 2656	. 2 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) )
24	23	ex 450	1 |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   `'ccnv 5113   dom cdm 5114   "cima 5117   Fun wfun 5882   -->wf 5884  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  frnfsuppbi  8304  frnnn0supp  11349  ffs2  29503  eulerpartlemmf  30437  pwfi2f1o  37666