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Metamath Proof Explorer

Theorem suppun

Description: The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019)

Ref Expression
Hypothesis suppun.g 
|- ( ph -> G e. V )
Assertion suppun 
|- ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )

Proof

Step Hyp Ref Expression
1 suppun.g 
 |-  ( ph -> G e. V )
2 ssun1 
 |-  ( `' F " ( _V \ { Z } ) ) C_ ( ( `' F " ( _V \ { Z } ) ) u. ( `' G " ( _V \ { Z } ) ) )
3 cnvun 
 |-  `' ( F u. G ) = ( `' F u. `' G )
4 3 imaeq1i 
 |-  ( `' ( F u. G ) " ( _V \ { Z } ) ) = ( ( `' F u. `' G ) " ( _V \ { Z } ) )
5 imaundir 
 |-  ( ( `' F u. `' G ) " ( _V \ { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' G " ( _V \ { Z } ) ) )
6 4 5 eqtri 
 |-  ( `' ( F u. G ) " ( _V \ { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' G " ( _V \ { Z } ) ) )
7 2 6 sseqtrri 
 |-  ( `' F " ( _V \ { Z } ) ) C_ ( `' ( F u. G ) " ( _V \ { Z } ) )
8 7 a1i 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( `' F " ( _V \ { Z } ) ) C_ ( `' ( F u. G ) " ( _V \ { Z } ) ) )
9 suppimacnv 
 |-  ( ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
10 9 adantr 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
11 unexg 
 |-  ( ( F e. _V /\ G e. V ) -> ( F u. G ) e. _V )
12 11 adantlr 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ G e. V ) -> ( F u. G ) e. _V )
13 1 12 sylan2 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( F u. G ) e. _V )
14 simplr 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> Z e. _V )
15 suppimacnv 
 |-  ( ( ( F u. G ) e. _V /\ Z e. _V ) -> ( ( F u. G ) supp Z ) = ( `' ( F u. G ) " ( _V \ { Z } ) ) )
16 13 14 15 syl2anc 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( ( F u. G ) supp Z ) = ( `' ( F u. G ) " ( _V \ { Z } ) ) )
17 8 10 16 3sstr4d 
 |-  ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )
18 17 ex 
 |-  ( ( F e. _V /\ Z e. _V ) -> ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) ) )
19 supp0prc 
 |-  ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = (/) )
20 0ss 
 |-  (/) C_ ( ( F u. G ) supp Z )
21 19 20 eqsstrdi 
 |-  ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )
22 21 a1d 
 |-  ( -. ( F e. _V /\ Z e. _V ) -> ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) ) )
23 18 22 pm2.61i 
 |-  ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )