Theorem fsuppeq 6425

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

Proof of Theorem fsuppeq

Step	Hyp	Ref	Expression
1		ffn 5489	. . . . 5 |- ( F : I --> S -> F Fn I )
2	1	adantl 277	. . . 4 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> F Fn I )
3		fex 5893	. . . . . . 7 |- ( ( F : I --> S /\ I e. V ) -> F e. _V )
4	3	expcom 116	. . . . . 6 |- ( I e. V -> ( F : I --> S -> F e. _V ) )
5	4	adantr 276	. . . . 5 |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> F e. _V ) )
6	5	imp 124	. . . 4 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> F e. _V )
7		simplr 529	. . . 4 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> Z e. W )
8		suppimacnvfn 6424	. . . 4 |- ( ( F Fn I /\ F e. _V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
9	2, 6, 7, 8	syl3anc 1274	. . 3 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
10		ffun 5492	. . . . . . 7 |- ( F : I --> S -> Fun F )
11		inpreima 5781	. . . . . . 7 |- ( Fun F -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) )
12	10, 11	syl 14	. . . . . 6 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) )
13		cnvimass 5106	. . . . . . . 8 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
14		fdm 5495	. . . . . . . . 9 |- ( F : I --> S -> dom F = I )
15		fimacnv 5784	. . . . . . . . 9 |- ( F : I --> S -> ( `' F " S ) = I )
16	14, 15	eqtr4d 2267	. . . . . . . 8 |- ( F : I --> S -> dom F = ( `' F " S ) )
17	13, 16	sseqtrid 3278	. . . . . . 7 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) )
18		sseqin2 3428	. . . . . . 7 |- ( ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) <-> ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
19	17, 18	sylib 122	. . . . . 6 |- ( F : I --> S -> ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
20	12, 19	eqtrd 2264	. . . . 5 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
21		invdif 3451	. . . . . 6 |- ( S i^i ( _V \ { Z } ) ) = ( S \ { Z } )
22	21	imaeq2i 5080	. . . . 5 |- ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( S \ { Z } ) )
23	20, 22	eqtr3di 2279	. . . 4 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) = ( `' F " ( S \ { Z } ) ) )
24	23	adantl 277	. . 3 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> ( `' F " ( _V \ { Z } ) ) = ( `' F " ( S \ { Z } ) ) )
25	9, 24	eqtrd 2264	. 2 |- ( ( ( I e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) )
26	25	ex 115	1 |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   \ cdif 3198   i^i cin 3200   C_ wss 3201   {csn 3673   `'ccnv 4730   dom cdm 4731   "cima 4734   Fun wfun 5327   Fn wfn 5328   -->wf 5329  (class class class)co 6028   supp csupp 6413

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by: (None)