Metamath Proof Explorer

Theorem funsssuppss

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)

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

Proof

Step Hyp Ref Expression
1 funss 
 |-  ( F C_ G -> ( Fun G -> Fun F ) )
2 1 impcom 
 |-  ( ( Fun G /\ F C_ G ) -> Fun F )
3 2 funfnd 
 |-  ( ( Fun G /\ F C_ G ) -> F Fn dom F )
4 funfn 
 |-  ( Fun G <-> G Fn dom G )
5 4 biimpi 
 |-  ( Fun G -> G Fn dom G )
6 5 adantr 
 |-  ( ( Fun G /\ F C_ G ) -> G Fn dom G )
7 3 6 jca 
 |-  ( ( Fun G /\ F C_ G ) -> ( F Fn dom F /\ G Fn dom G ) )
8 7 3adant3 
 |-  ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F Fn dom F /\ G Fn dom G ) )
9 8 adantr 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( F Fn dom F /\ G Fn dom G ) )
10 dmss 
 |-  ( F C_ G -> dom F C_ dom G )
11 10 3ad2ant2 
 |-  ( ( Fun G /\ F C_ G /\ G e. V ) -> dom F C_ dom G )
12 11 adantr 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> dom F C_ dom G )
13 dmexg 
 |-  ( G e. V -> dom G e. _V )
14 13 3ad2ant3 
 |-  ( ( Fun G /\ F C_ G /\ G e. V ) -> dom G e. _V )
15 14 adantr 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> dom G e. _V )
16 simpr 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> Z e. _V )
17 12 15 16 3jca 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) )
18 9 17 jca 
 |-  ( ( ( 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 ) ) )
19 funssfv 
 |-  ( ( Fun G /\ F C_ G /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
20 19 3expa 
 |-  ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
21 eqeq1 
 |-  ( ( G ` x ) = ( F ` x ) -> ( ( G ` x ) = Z <-> ( F ` x ) = Z ) )
22 21 biimpd 
 |-  ( ( G ` x ) = ( F ` x ) -> ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
23 20 22 syl 
 |-  ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
24 23 ralrimiva 
 |-  ( ( Fun G /\ F C_ G ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
25 24 3adant3 
 |-  ( ( Fun G /\ F C_ G /\ G e. V ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
26 25 adantr 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
27 suppfnss 
 |-  ( ( ( 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 ) ) )
28 18 26 27 sylc 
 |-  ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( F supp Z ) C_ ( G supp Z ) )
29 28 expcom 
 |-  ( Z e. _V -> ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) ) )
30 ssid 
 |-  (/) C_ (/)
31 simpr 
 |-  ( ( F e. _V /\ Z e. _V ) -> Z e. _V )
32 supp0prc 
 |-  ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = (/) )
33 31 32 nsyl5 
 |-  ( -. Z e. _V -> ( F supp Z ) = (/) )
34 simpr 
 |-  ( ( G e. _V /\ Z e. _V ) -> Z e. _V )
35 supp0prc 
 |-  ( -. ( G e. _V /\ Z e. _V ) -> ( G supp Z ) = (/) )
36 34 35 nsyl5 
 |-  ( -. Z e. _V -> ( G supp Z ) = (/) )
37 33 36 sseq12d 
 |-  ( -. Z e. _V -> ( ( F supp Z ) C_ ( G supp Z ) <-> (/) C_ (/) ) )
38 30 37 mpbiri 
 |-  ( -. Z e. _V -> ( F supp Z ) C_ ( G supp Z ) )
39 38 a1d 
 |-  ( -. Z e. _V -> ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) ) )
40 29 39 pm2.61i 
 |-  ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )