Theorem suppssdc 6459

Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)

Hypotheses

Ref	Expression
suppss.f	|- ( ph -> F : A --> B )
suppss.n	|- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )
suppssdc.dc	|- ( ph -> A. x e. A DECID x e. W )

Assertion

Ref	Expression
suppssdc	|- ( ph -> ( F supp Z ) C_ W )

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

Allowed substitution hints:   ph( x)   A( k)   B( x, k)   F( x)   Z( x)

Proof of Theorem suppssdc

Dummy variables f i z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-supp 6435	. . . 4 |- supp = ( f e. _V , z e. _V |-> { i e. dom f | ( f " { i } ) =/= { z } } )
2	1	elmpocl 6248	. . 3 |- ( k e. ( F supp Z ) -> ( F e. _V /\ Z e. _V ) )
3		suppss.f	. . . . . . . . 9 |- ( ph -> F : A --> B )
4	3	ffnd 5508	. . . . . . . 8 |- ( ph -> F Fn A )
5	4	adantl 277	. . . . . . 7 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> F Fn A )
6		simpll 527	. . . . . . 7 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> F e. _V )
7		simplr 529	. . . . . . 7 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> Z e. _V )
8		elsuppfng 6441	. . . . . . 7 |- ( ( F Fn A /\ F e. _V /\ Z e. _V ) -> ( k e. ( F supp Z ) <-> ( k e. A /\ ( F ` k ) =/= Z ) ) )
9	5, 6, 7, 8	syl3anc 1274	. . . . . 6 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( k e. ( F supp Z ) <-> ( k e. A /\ ( F ` k ) =/= Z ) ) )
10		eleq1w 2293	. . . . . . . . . 10 |- ( x = k -> ( x e. W <-> k e. W ) )
11	10	dcbid 846	. . . . . . . . 9 |- ( x = k -> (DECID x e. W <-> DECID k e. W ) )
12		suppssdc.dc	. . . . . . . . . 10 |- ( ph -> A. x e. A DECID x e. W )
13	12	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> A. x e. A DECID x e. W )
14		simpr 110	. . . . . . . . 9 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> k e. A )
15	11, 13, 14	rspcdva 2925	. . . . . . . 8 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> DECID k e. W )
16		eldif 3219	. . . . . . . . . . . 12 |- ( k e. ( A \ W ) <-> ( k e. A /\ -. k e. W ) )
17		suppss.n	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )
18	17	adantll 476	. . . . . . . . . . . 12 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )
19	16, 18	sylan2br 288	. . . . . . . . . . 11 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ ( k e. A /\ -. k e. W ) ) -> ( F ` k ) = Z )
20	19	expr 375	. . . . . . . . . 10 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> ( -. k e. W -> ( F ` k ) = Z ) )
21	20	a1d 22	. . . . . . . . 9 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> (DECID k e. W -> ( -. k e. W -> ( F ` k ) = Z ) ) )
22	21	necon1addc 2488	. . . . . . . 8 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> (DECID k e. W -> ( ( F ` k ) =/= Z -> k e. W ) ) )
23	15, 22	mpd 13	. . . . . . 7 |- ( ( ( ( F e. _V /\ Z e. _V ) /\ ph ) /\ k e. A ) -> ( ( F ` k ) =/= Z -> k e. W ) )
24	23	expimpd 363	. . . . . 6 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( ( k e. A /\ ( F ` k ) =/= Z ) -> k e. W ) )
25	9, 24	sylbid 150	. . . . 5 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( k e. ( F supp Z ) -> k e. W ) )
26	25	expcom 116	. . . 4 |- ( ph -> ( ( F e. _V /\ Z e. _V ) -> ( k e. ( F supp Z ) -> k e. W ) ) )
27	26	com23 78	. . 3 |- ( ph -> ( k e. ( F supp Z ) -> ( ( F e. _V /\ Z e. _V ) -> k e. W ) ) )
28	2, 27	mpdi 43	. 2 |- ( ph -> ( k e. ( F supp Z ) -> k e. W ) )
29	28	ssrdv 3243	1 |- ( ph -> ( F supp Z ) C_ W )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   {crab 2524   _Vcvv 2812   \ cdif 3207   C_ wss 3210   {csn 3688   dom cdm 4748   "cima 4751   Fn wfn 5346   -->wf 5347   ` 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-stab 839  df-dc 843  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-f 5355  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by: (None)