Theorem fsuppeq 6460

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 5513	. . . . 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 5920	. . . . . . 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 6459	. . . 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 5516	. . . . . . 7 |- ( F : I --> S -> Fun F )
11		inpreima 5808	. . . . . . 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 5130	. . . . . . . 8 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
14		fdm 5519	. . . . . . . . 9 |- ( F : I --> S -> dom F = I )
15		fimacnv 5811	. . . . . . . . 9 |- ( F : I --> S -> ( `' F " S ) = I )
16	14, 15	eqtr4d 2270	. . . . . . . 8 |- ( F : I --> S -> dom F = ( `' F " S ) )
17	13, 16	sseqtrid 3292	. . . . . . 7 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) )
18		sseqin2 3444	. . . . . . 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 2267	. . . . 5 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
21		invdif 3467	. . . . . 6 |- ( S i^i ( _V \ { Z } ) ) = ( S \ { Z } )
22	21	imaeq2i 5104	. . . . 5 |- ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( S \ { Z } ) )
23	20, 22	eqtr3di 2282	. . . 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 2267	. 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 2205   _Vcvv 2815   \ cdif 3211   i^i cin 3213   C_ wss 3214   {csn 3694   `'ccnv 4753   dom cdm 4754   "cima 4757   Fun wfun 5351   Fn wfn 5352   -->wf 5353  (class class class)co 6058   supp csupp 6448

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by:  ffsuppbi  7266  fcdmnn0supp  9565