Theorem suppssOLD 5996

Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6922 as of 28-May-2019. (New usage is discouraged.) (Proof modification 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 5713	. . . 4 |- ( F : A --> B -> F Fn A )
3		elpreima 5983	. . . 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 5858	. . . . . 6 |- ( F ` k ) e. _V
6		eldifsn 4141	. . . . . 6 |- ( ( F ` k ) e. ( _V \ { Z } ) <-> ( ( F ` k ) e. _V /\ ( F ` k ) =/= Z ) )
7	5, 6	mpbiran 916	. . . . 5 |- ( ( F ` k ) e. ( _V \ { Z } ) <-> ( F ` k ) =/= Z )
8		eldif 3471	. . . . . . . 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 474	. . . . . . 7 |- ( ( ph /\ ( k e. A /\ -. k e. W ) ) -> ( F ` k ) = Z )
11	10	expr 613	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( -. k e. W -> ( F ` k ) = Z ) )
12	11	necon1ad 2670	. . . . 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 601	. . 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 3495	1 |- ( ph -> ( `' F " ( _V \ { Z } ) ) C_ W )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   _Vcvv 3106   \ cdif 3458   C_ wss 3461   {csn 4016   `'ccnv 4987   "cima 4991   Fn wfn 5565   -->wf 5566   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578

This theorem is referenced by:  cantnfp1lem1OLD  8114  cantnfp1lem3OLD  8116  gsumzaddlemOLD  17135  gsumzmhmOLD  17156  gsumzinvOLD  17168  lcomfsupOLD  17744  psrbaglesuppOLD  18213  psrlidmOLD  18252  psrridmOLD  18254  mplsubglemOLD  18290  mpllsslemOLD  18291  mplsubrglemOLD  18296