Theorem funsssuppss 7321

Description: The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)

Assertion

Ref	Expression
funsssuppss	|- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )

Proof of Theorem funsssuppss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funss 5907	. . . . . . . . . 10 |- ( F C_ G -> ( Fun G -> Fun F ) )
2	1	impcom 446	. . . . . . . . 9 |- ( ( Fun G /\ F C_ G ) -> Fun F )
3		funfn 5918	. . . . . . . . . 10 |- ( Fun F <-> F Fn dom F )
4	3	biimpi 206	. . . . . . . . 9 |- ( Fun F -> F Fn dom F )
5	2, 4	syl 17	. . . . . . . 8 |- ( ( Fun G /\ F C_ G ) -> F Fn dom F )
6		funfn 5918	. . . . . . . . . 10 |- ( Fun G <-> G Fn dom G )
7	6	biimpi 206	. . . . . . . . 9 |- ( Fun G -> G Fn dom G )
8	7	adantr 481	. . . . . . . 8 |- ( ( Fun G /\ F C_ G ) -> G Fn dom G )
9	5, 8	jca 554	. . . . . . 7 |- ( ( Fun G /\ F C_ G ) -> ( F Fn dom F /\ G Fn dom G ) )
10	9	3adant3 1081	. . . . . 6 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F Fn dom F /\ G Fn dom G ) )
11	10	adantr 481	. . . . 5 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( F Fn dom F /\ G Fn dom G ) )
12		dmss 5323	. . . . . . . 8 |- ( F C_ G -> dom F C_ dom G )
13	12	3ad2ant2 1083	. . . . . . 7 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> dom F C_ dom G )
14	13	adantr 481	. . . . . 6 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> dom F C_ dom G )
15		dmexg 7097	. . . . . . . 8 |- ( G e. V -> dom G e. _V )
16	15	3ad2ant3 1084	. . . . . . 7 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> dom G e. _V )
17	16	adantr 481	. . . . . 6 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> dom G e. _V )
18		simpr 477	. . . . . 6 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> Z e. _V )
19	14, 17, 18	3jca 1242	. . . . 5 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) )
20	11, 19	jca 554	. . . 4 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( ( F Fn dom F /\ G Fn dom G ) /\ ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) ) )
21		funssfv 6209	. . . . . . . . 9 |- ( ( Fun G /\ F C_ G /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
22	21	3expa 1265	. . . . . . . 8 |- ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
23		eqeq1 2626	. . . . . . . . 9 |- ( ( G ` x ) = ( F ` x ) -> ( ( G ` x ) = Z <-> ( F ` x ) = Z ) )
24	23	biimpd 219	. . . . . . . 8 |- ( ( G ` x ) = ( F ` x ) -> ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
25	22, 24	syl 17	. . . . . . 7 |- ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
26	25	ralrimiva 2966	. . . . . 6 |- ( ( Fun G /\ F C_ G ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
27	26	3adant3 1081	. . . . 5 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
28	27	adantr 481	. . . 4 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
29		suppfnss 7320	. . . 4 |- ( ( ( F Fn dom F /\ G Fn dom G ) /\ ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) ) -> ( A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( F supp Z ) C_ ( G supp Z ) ) )
30	20, 28, 29	sylc 65	. . 3 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( F supp Z ) C_ ( G supp Z ) )
31	30	expcom 451	. 2 |- ( Z e. _V -> ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) ) )
32		ssid 3624	. . . 4 |- (/) C_ (/)
33		simpr 477	. . . . . . 7 |- ( ( F e. _V /\ Z e. _V ) -> Z e. _V )
34	33	con3i 150	. . . . . 6 |- ( -. Z e. _V -> -. ( F e. _V /\ Z e. _V ) )
35		supp0prc 7298	. . . . . 6 |- ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = (/) )
36	34, 35	syl 17	. . . . 5 |- ( -. Z e. _V -> ( F supp Z ) = (/) )
37		simpr 477	. . . . . . 7 |- ( ( G e. _V /\ Z e. _V ) -> Z e. _V )
38	37	con3i 150	. . . . . 6 |- ( -. Z e. _V -> -. ( G e. _V /\ Z e. _V ) )
39		supp0prc 7298	. . . . . 6 |- ( -. ( G e. _V /\ Z e. _V ) -> ( G supp Z ) = (/) )
40	38, 39	syl 17	. . . . 5 |- ( -. Z e. _V -> ( G supp Z ) = (/) )
41	36, 40	sseq12d 3634	. . . 4 |- ( -. Z e. _V -> ( ( F supp Z ) C_ ( G supp Z ) <-> (/) C_ (/) ) )
42	32, 41	mpbiri 248	. . 3 |- ( -. Z e. _V -> ( F supp Z ) C_ ( G supp Z ) )
43	42	a1d 25	. 2 |- ( -. Z e. _V -> ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) ) )
44	31, 43	pm2.61i 176	1 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (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:  tdeglem4  23820