Theorem suppimacnvfn 6445

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 1025	. . . . . . . . 9 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> F e. V )
2		vex 2815	. . . . . . . . 9 |- x e. _V
3		fvexg 5688	. . . . . . . . 9 |- ( ( F e. V /\ x e. _V ) -> ( F ` x ) e. _V )
4	1, 2, 3	sylancl 413	. . . . . . . 8 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F ` x ) e. _V )
5		elsng 3703	. . . . . . . 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 2452	. . . . . 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 3219	. . . . 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 464	. . 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 6441	. . 3 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( x e. ( F supp Z ) <-> ( x e. X /\ ( F ` x ) =/= Z ) ) )
14		elpreima 5796	. . . 4 |- ( F Fn X -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. X /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
15	14	3ad2ant1 1045	. . 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 2230	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 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   _Vcvv 2812   \ cdif 3207   {csn 3688   `'ccnv 4747   "cima 4751   Fn wfn 5346   ` cfv 5351  (class class class)co 6049   supp csupp 6434

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by:  fsuppeq  6446  fsuppeqg  6447  mptsuppdifd  6454  suppcofn  6465