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Theorem suppun 6910
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.)
Hypothesis
Ref Expression
suppun.g  |-  ( ph  ->  G  e.  V )
Assertion
Ref Expression
suppun  |-  ( ph  ->  ( F supp  Z ) 
C_  ( ( F  u.  G ) supp  Z
) )

Proof of Theorem suppun
StepHypRef Expression
1 ssun1 3660 . . . . . 6  |-  ( `' F " ( _V 
\  { Z }
) )  C_  (
( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
2 cnvun 5402 . . . . . . . 8  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
32imaeq1i 5325 . . . . . . 7  |-  ( `' ( F  u.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' F  u.  `' G ) " ( _V  \  { Z }
) )
4 imaundir 5410 . . . . . . 7  |-  ( ( `' F  u.  `' G ) " ( _V  \  { Z }
) )  =  ( ( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
53, 4eqtri 2489 . . . . . 6  |-  ( `' ( F  u.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
61, 5sseqtr4i 3530 . . . . 5  |-  ( `' F " ( _V 
\  { Z }
) )  C_  ( `' ( F  u.  G ) " ( _V  \  { Z }
) )
76a1i 11 . . . 4  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( `' F " ( _V 
\  { Z }
) )  C_  ( `' ( F  u.  G ) " ( _V  \  { Z }
) ) )
8 suppimacnv 6902 . . . . 5  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
98adantr 465 . . . 4  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F supp  Z )  =  ( `' F " ( _V 
\  { Z }
) ) )
10 suppun.g . . . . . 6  |-  ( ph  ->  G  e.  V )
11 unexg 6576 . . . . . . 7  |-  ( ( F  e.  _V  /\  G  e.  V )  ->  ( F  u.  G
)  e.  _V )
1211adantlr 714 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  G  e.  V
)  ->  ( F  u.  G )  e.  _V )
1310, 12sylan2 474 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F  u.  G )  e.  _V )
14 simplr 754 . . . . 5  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  Z  e.  _V )
15 suppimacnv 6902 . . . . 5  |-  ( ( ( F  u.  G
)  e.  _V  /\  Z  e.  _V )  ->  ( ( F  u.  G ) supp  Z )  =  ( `' ( F  u.  G )
" ( _V  \  { Z } ) ) )
1613, 14, 15syl2anc 661 . . . 4  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
( F  u.  G
) supp  Z )  =  ( `' ( F  u.  G ) " ( _V  \  { Z }
) ) )
177, 9, 163sstr4d 3540 . . 3  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F supp  Z )  C_  (
( F  u.  G
) supp  Z ) )
1817ex 434 . 2  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  ( ph  ->  ( F supp  Z )  C_  (
( F  u.  G
) supp  Z ) ) )
19 supp0prc 6894 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
20 0ss 3807 . . . 4  |-  (/)  C_  (
( F  u.  G
) supp  Z )
2119, 20syl6eqss 3547 . . 3  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z ) 
C_  ( ( F  u.  G ) supp  Z
) )
2221a1d 25 . 2  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( ph  ->  ( F supp  Z )  C_  (
( F  u.  G
) supp  Z ) ) )
2318, 22pm2.61i 164 1  |-  ( ph  ->  ( F supp  Z ) 
C_  ( ( F  u.  G ) supp  Z
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    u. cun 3467    C_ wss 3469   (/)c0 3778   {csn 4020   `'ccnv 4991   "cima 4995  (class class class)co 6275   supp csupp 6891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-supp 6892
This theorem is referenced by:  fsuppunbi  7839  gsumzaddlem  16718
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