Theorem suppimacnvfn 6480

Description: Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)

Assertion

Ref	Expression
suppimacnvfn	|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )

Proof of Theorem suppimacnvfn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1029	. . . . . . . . 9 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> F e. V )
2		vex 2824	. . . . . . . . 9 |- x e. _V
3		fvexg 5712	. . . . . . . . 9 |- ( ( F e. V /\ x e. _V ) -> ( F ` x ) e. _V )
4	1, 2, 3	sylancl 417	. . . . . . . 8 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F ` x ) e. _V )
5		elsng 3723	. . . . . . . 8 |- ( ( F ` x ) e. _V -> ( ( F ` x ) e. { Z } <-> ( F ` x ) = Z ) )
6	4, 5	syl 14	. . . . . . 7 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( ( F ` x ) e. { Z } <-> ( F ` x ) = Z ) )
7	6	necon3bbid 2460	. . . . . 6 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( -. ( F ` x ) e. { Z } <-> ( F ` x ) =/= Z ) )
8	4	biantrurd 305	. . . . . 6 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( -. ( F ` x ) e. { Z } <-> ( ( F ` x ) e. _V /\ -. ( F ` x ) e. { Z } ) ) )
9	7, 8	bitr3d 190	. . . . 5 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( ( F ` x ) =/= Z <-> ( ( F ` x ) e. _V /\ -. ( F ` x ) e. { Z } ) ) )
10		eldif 3229	. . . . 5 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( ( F ` x ) e. _V /\ -. ( F ` x ) e. { Z } ) )
11	9, 10	bitr4di 198	. . . 4 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( ( F ` x ) =/= Z <-> ( F ` x ) e. ( _V \ { Z } ) ) )
12	11	anbi2d 468	. . 3 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( ( x e. X /\ ( F ` x ) =/= Z ) <-> ( x e. X /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
13		elsuppfng 6476	. . 3 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( x e. ( F supp Z ) <-> ( x e. X /\ ( F ` x ) =/= Z ) ) )
14		elpreima 5822	. . . 4 |- ( F Fn X -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. X /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
15	14	3ad2ant1 1049	. . 3 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. X /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
16	12, 13, 15	3bitr4d 220	. 2 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( x e. ( F supp Z ) <-> x e. ( `' F " ( _V \ { Z } ) ) ) )
17	16	eqrdv 2236	1 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   \ cdif 3217   {csn 3708   `'ccnv 4771   "cima 4775   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   supp csupp 6469

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  fsuppeq  6481  fsuppeqg  6482  mptsuppdifd  6489  suppcofn  6500