Theorem suppss2 7329

Description: Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.)

Hypotheses

Ref	Expression
suppss2.n	|- ( ( ph /\ k e. ( A \ W ) ) -> B = Z )
suppss2.a	|- ( ph -> A e. V )

Assertion

Ref	Expression
suppss2	|- ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W )

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

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

Proof of Theorem suppss2

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( k e. A |-> B ) = ( k e. A |-> B )
2		suppss2.a	. . . . . 6 |- ( ph -> A e. V )
3	2	adantl 482	. . . . 5 |- ( ( Z e. _V /\ ph ) -> A e. V )
4		simpl 473	. . . . 5 |- ( ( Z e. _V /\ ph ) -> Z e. _V )
5	1, 3, 4	mptsuppdifd 7317	. . . 4 |- ( ( Z e. _V /\ ph ) -> ( ( k e. A |-> B ) supp Z ) = { k e. A | B e. ( _V \ { Z } ) } )
6		eldifsni 4320	. . . . . . 7 |- ( B e. ( _V \ { Z } ) -> B =/= Z )
7		eldif 3584	. . . . . . . . . 10 |- ( k e. ( A \ W ) <-> ( k e. A /\ -. k e. W ) )
8		suppss2.n	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( A \ W ) ) -> B = Z )
9	8	adantll 750	. . . . . . . . . 10 |- ( ( ( Z e. _V /\ ph ) /\ k e. ( A \ W ) ) -> B = Z )
10	7, 9	sylan2br 493	. . . . . . . . 9 |- ( ( ( Z e. _V /\ ph ) /\ ( k e. A /\ -. k e. W ) ) -> B = Z )
11	10	expr 643	. . . . . . . 8 |- ( ( ( Z e. _V /\ ph ) /\ k e. A ) -> ( -. k e. W -> B = Z ) )
12	11	necon1ad 2811	. . . . . . 7 |- ( ( ( Z e. _V /\ ph ) /\ k e. A ) -> ( B =/= Z -> k e. W ) )
13	6, 12	syl5 34	. . . . . 6 |- ( ( ( Z e. _V /\ ph ) /\ k e. A ) -> ( B e. ( _V \ { Z } ) -> k e. W ) )
14	13	3impia 1261	. . . . 5 |- ( ( ( Z e. _V /\ ph ) /\ k e. A /\ B e. ( _V \ { Z } ) ) -> k e. W )
15	14	rabssdv 3682	. . . 4 |- ( ( Z e. _V /\ ph ) -> { k e. A | B e. ( _V \ { Z } ) } C_ W )
16	5, 15	eqsstrd 3639	. . 3 |- ( ( Z e. _V /\ ph ) -> ( ( k e. A |-> B ) supp Z ) C_ W )
17	16	ex 450	. 2 |- ( Z e. _V -> ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W ) )
18		id 22	. . . . . 6 |- ( -. Z e. _V -> -. Z e. _V )
19	18	intnand 962	. . . . 5 |- ( -. Z e. _V -> -. ( ( k e. A |-> B ) e. _V /\ Z e. _V ) )
20		supp0prc 7298	. . . . 5 |- ( -. ( ( k e. A |-> B ) e. _V /\ Z e. _V ) -> ( ( k e. A |-> B ) supp Z ) = (/) )
21	19, 20	syl 17	. . . 4 |- ( -. Z e. _V -> ( ( k e. A |-> B ) supp Z ) = (/) )
22		0ss 3972	. . . 4 |- (/) C_ W
23	21, 22	syl6eqss 3655	. . 3 |- ( -. Z e. _V -> ( ( k e. A |-> B ) supp Z ) C_ W )
24	23	a1d 25	. 2 |- ( -. Z e. _V -> ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W ) )
25	17, 24	pm2.61i 176	1 |- ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  suppsssn  7330  fsuppmptif  8305  sniffsupp  8315  cantnflem1d  8585  cantnflem1  8586  gsumzsplit  18327  gsummpt1n0  18364  gsum2dlem1  18369  gsum2dlem2  18370  gsum2d  18371  dprdfid  18416  dprdfinv  18418  dprdfadd  18419  dmdprdsplitlem  18436  dpjidcl  18457  psrbagaddcl  19370  psrlidm  19403  psrridm  19404  mplsubrg  19440  mplmon  19463  mplmonmul  19464  mplcoe1  19465  mplcoe5  19468  mplbas2  19470  evlslem4  19508  evlslem2  19512  evlslem3  19514  evlslem1  19515  coe1tmmul2  19646  coe1tmmul  19647  uvcff  20130  uvcresum  20132  tsmssplit  21955  coe1mul3  23859  plypf1  23968  tayl0  24116  suppss2f  29439  suppss3  29502