Theorem suppssOLD 5932

Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6816 as of 28-May-2019. (New usage is discouraged.)

Hypotheses

Ref	Expression
suppssOLD.f	|- ( ph -> F : A --> B )
suppssOLD.n	|- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )

Assertion

Ref	Expression
suppssOLD	|- ( ph -> ( `' F " ( _V \ { Z } ) ) C_ W )

Distinct variable groups:   k, F   ph, k   k, W   k, Z

Allowed substitution hints:   A( k)   B( k)

Proof of Theorem suppssOLD

Step	Hyp	Ref	Expression
1		suppssOLD.f	. . . 4 |- ( ph -> F : A --> B )
2		ffn 5654	. . . 4 |- ( F : A --> B -> F Fn A )
3		elpreima 5919	. . . 4 |- ( F Fn A -> ( k e. ( `' F " ( _V \ { Z } ) ) <-> ( k e. A /\ ( F ` k ) e. ( _V \ { Z } ) ) ) )
4	1, 2, 3	3syl 20	. . 3 |- ( ph -> ( k e. ( `' F " ( _V \ { Z } ) ) <-> ( k e. A /\ ( F ` k ) e. ( _V \ { Z } ) ) ) )
5		fvex 5796	. . . . . 6 |- ( F ` k ) e. _V
6		eldifsn 4095	. . . . . 6 |- ( ( F ` k ) e. ( _V \ { Z } ) <-> ( ( F ` k ) e. _V /\ ( F ` k ) =/= Z ) )
7	5, 6	mpbiran 909	. . . . 5 |- ( ( F ` k ) e. ( _V \ { Z } ) <-> ( F ` k ) =/= Z )
8		eldif 3433	. . . . . . . 8 |- ( k e. ( A \ W ) <-> ( k e. A /\ -. k e. W ) )
9		suppssOLD.n	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )
10	8, 9	sylan2br 476	. . . . . . 7 |- ( ( ph /\ ( k e. A /\ -. k e. W ) ) -> ( F ` k ) = Z )
11	10	expr 615	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( -. k e. W -> ( F ` k ) = Z ) )
12	11	necon1ad 2662	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) =/= Z -> k e. W ) )
13	7, 12	syl5bi 217	. . . 4 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) e. ( _V \ { Z } ) -> k e. W ) )
14	13	expimpd 603	. . 3 |- ( ph -> ( ( k e. A /\ ( F ` k ) e. ( _V \ { Z } ) ) -> k e. W ) )
15	4, 14	sylbid 215	. 2 |- ( ph -> ( k e. ( `' F " ( _V \ { Z } ) ) -> k e. W ) )
16	15	ssrdv 3457	1 |- ( ph -> ( `' F " ( _V \ { Z } ) ) C_ W )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2642   _Vcvv 3065   \ cdif 3420   C_ wss 3423   {csn 3972   `'ccnv 4934   "cima 4938   Fn wfn 5508   -->wf 5509   ` cfv 5513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521

This theorem is referenced by:  cantnfp1lem1OLD  8010  cantnfp1lem3OLD  8012  gsumzaddlemOLD  16511  gsumzmhmOLD  16533  gsumzinvOLD  16545  gsumsubOLD  16550  lcomfsupOLD  17087  psrbaglesuppOLD  17539  psrlidmOLD  17578  psrridmOLD  17580  mplsubglemOLD  17616  mpllsslemOLD  17617  mplsubrglemOLD  17622  frlmssuvc1OLD  18327  frlmsslspOLD  18330