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Theorem suppun 6820
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 3628 . . . . . 6  |-  ( `' F " ( _V 
\  { Z }
) )  C_  (
( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
2 cnvun 5351 . . . . . . . 8  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
32imaeq1i 5275 . . . . . . 7  |-  ( `' ( F  u.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' F  u.  `' G ) " ( _V  \  { Z }
) )
4 imaundir 5359 . . . . . . 7  |-  ( ( `' F  u.  `' G ) " ( _V  \  { Z }
) )  =  ( ( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
53, 4eqtri 2483 . . . . . 6  |-  ( `' ( F  u.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
61, 5sseqtr4i 3498 . . . . 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 6812 . . . . 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 6492 . . . . . . 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 6812 . . . . 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 3508 . . 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 6804 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
20 0ss 3775 . . . 4  |-  (/)  C_  (
( F  u.  G
) supp  Z )
2119, 20syl6eqss 3515 . . 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 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3434    u. cun 3435    C_ wss 3437   (/)c0 3746   {csn 3986   `'ccnv 4948   "cima 4952  (class class class)co 6201   supp csupp 6801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-supp 6802
This theorem is referenced by:  fsuppunbi  7753  gsumzaddlem  16530
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