Theorem suppss2f 29439

Description: Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)

Hypotheses

Ref	Expression
suppss2f.p	|- F/ k ph
suppss2f.a	|- F/_ k A
suppss2f.w	|- F/_ k W
suppss2f.n	|- ( ( ph /\ k e. ( A \ W ) ) -> B = Z )
suppss2f.v	|- ( ph -> A e. V )

Assertion

Ref	Expression
suppss2f	|- ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W )

Distinct variable group:   k, Z

Allowed substitution hints:   ph( k)   A( k)   B( k)   V( k)   W( k)

Proof of Theorem suppss2f

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppss2f.a	. . . 4 |- F/_ k A
2		nfcv 2764	. . . 4 |- F/_ l A
3		nfcv 2764	. . . 4 |- F/_ l B
4		nfcsb1v 3549	. . . 4 |- F/_ k [_ l / k ]_ B
5		csbeq1a 3542	. . . 4 |- ( k = l -> B = [_ l / k ]_ B )
6	1, 2, 3, 4, 5	cbvmptf 4748	. . 3 |- ( k e. A |-> B ) = ( l e. A |-> [_ l / k ]_ B )
7	6	oveq1i 6660	. 2 |- ( ( k e. A |-> B ) supp Z ) = ( ( l e. A |-> [_ l / k ]_ B ) supp Z )
8		suppss2f.n	. . . . 5 |- ( ( ph /\ k e. ( A \ W ) ) -> B = Z )
9	8	sbt 2419	. . . 4 |- [ l / k ] ( ( ph /\ k e. ( A \ W ) ) -> B = Z )
10		sbim 2395	. . . . 5 |- ( [ l / k ] ( ( ph /\ k e. ( A \ W ) ) -> B = Z ) <-> ( [ l / k ] ( ph /\ k e. ( A \ W ) ) -> [ l / k ] B = Z ) )
11		sban 2399	. . . . . . 7 |- ( [ l / k ] ( ph /\ k e. ( A \ W ) ) <-> ( [ l / k ] ph /\ [ l / k ] k e. ( A \ W ) ) )
12		suppss2f.p	. . . . . . . . 9 |- F/ k ph
13	12	sbf 2380	. . . . . . . 8 |- ( [ l / k ] ph <-> ph )
14		suppss2f.w	. . . . . . . . . 10 |- F/_ k W
15	1, 14	nfdif 3731	. . . . . . . . 9 |- F/_ k ( A \ W )
16	15	clelsb3f 2768	. . . . . . . 8 |- ( [ l / k ] k e. ( A \ W ) <-> l e. ( A \ W ) )
17	13, 16	anbi12i 733	. . . . . . 7 |- ( ( [ l / k ] ph /\ [ l / k ] k e. ( A \ W ) ) <-> ( ph /\ l e. ( A \ W ) ) )
18	11, 17	bitri 264	. . . . . 6 |- ( [ l / k ] ( ph /\ k e. ( A \ W ) ) <-> ( ph /\ l e. ( A \ W ) ) )
19		sbsbc 3439	. . . . . . 7 |- ( [ l / k ] B = Z <-> [. l / k ]. B = Z )
20		vex 3203	. . . . . . . 8 |- l e. _V
21		sbceq1g 3988	. . . . . . . 8 |- ( l e. _V -> ( [. l / k ]. B = Z <-> [_ l / k ]_ B = Z ) )
22	20, 21	ax-mp 5	. . . . . . 7 |- ( [. l / k ]. B = Z <-> [_ l / k ]_ B = Z )
23	19, 22	bitri 264	. . . . . 6 |- ( [ l / k ] B = Z <-> [_ l / k ]_ B = Z )
24	18, 23	imbi12i 340	. . . . 5 |- ( ( [ l / k ] ( ph /\ k e. ( A \ W ) ) -> [ l / k ] B = Z ) <-> ( ( ph /\ l e. ( A \ W ) ) -> [_ l / k ]_ B = Z ) )
25	10, 24	bitri 264	. . . 4 |- ( [ l / k ] ( ( ph /\ k e. ( A \ W ) ) -> B = Z ) <-> ( ( ph /\ l e. ( A \ W ) ) -> [_ l / k ]_ B = Z ) )
26	9, 25	mpbi 220	. . 3 |- ( ( ph /\ l e. ( A \ W ) ) -> [_ l / k ]_ B = Z )
27		suppss2f.v	. . 3 |- ( ph -> A e. V )
28	26, 27	suppss2 7329	. 2 |- ( ph -> ( ( l e. A |-> [_ l / k ]_ B ) supp Z ) C_ W )
29	7, 28	syl5eqss 3649	1 |- ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435   [_csb 3533   \ cdif 3571   C_ wss 3574   |-> 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:  esumss  30134