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Theorem suppun 6875
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 3603 . . . . . 6  |-  ( `' F " ( _V 
\  { Z }
) )  C_  (
( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
2 cnvun 5348 . . . . . . . 8  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
32imaeq1i 5273 . . . . . . 7  |-  ( `' ( F  u.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' F  u.  `' G ) " ( _V  \  { Z }
) )
4 imaundir 5356 . . . . . . 7  |-  ( ( `' F  u.  `' G ) " ( _V  \  { Z }
) )  =  ( ( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
53, 4eqtri 2429 . . . . . 6  |-  ( `' ( F  u.  G
) " ( _V 
\  { Z }
) )  =  ( ( `' F "
( _V  \  { Z } ) )  u.  ( `' G "
( _V  \  { Z } ) ) )
61, 5sseqtr4i 3472 . . . . 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 6865 . . . . 5  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
98adantr 463 . . . 4  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F supp  Z )  =  ( `' F " ( _V 
\  { Z }
) ) )
10 suppun.g . . . . . 6  |-  ( ph  ->  G  e.  V )
11 unexg 6537 . . . . . . 7  |-  ( ( F  e.  _V  /\  G  e.  V )  ->  ( F  u.  G
)  e.  _V )
1211adantlr 713 . . . . . 6  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  G  e.  V
)  ->  ( F  u.  G )  e.  _V )
1310, 12sylan2 472 . . . . 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 6865 . . . . 5  |-  ( ( ( F  u.  G
)  e.  _V  /\  Z  e.  _V )  ->  ( ( F  u.  G ) supp  Z )  =  ( `' ( F  u.  G )
" ( _V  \  { Z } ) ) )
1613, 14, 15syl2anc 659 . . . 4  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  (
( F  u.  G
) supp  Z )  =  ( `' ( F  u.  G ) " ( _V  \  { Z }
) ) )
177, 9, 163sstr4d 3482 . . 3  |-  ( ( ( F  e.  _V  /\  Z  e.  _V )  /\  ph )  ->  ( F supp  Z )  C_  (
( F  u.  G
) supp  Z ) )
1817ex 432 . 2  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  ( ph  ->  ( F supp  Z )  C_  (
( F  u.  G
) supp  Z ) ) )
19 supp0prc 6857 . . . 4  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
20 0ss 3765 . . . 4  |-  (/)  C_  (
( F  u.  G
) supp  Z )
2119, 20syl6eqss 3489 . . 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 367    = wceq 1403    e. wcel 1840   _Vcvv 3056    \ cdif 3408    u. cun 3409    C_ wss 3411   (/)c0 3735   {csn 3969   `'ccnv 4939   "cima 4943  (class class class)co 6232   supp csupp 6854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-supp 6855
This theorem is referenced by:  fsuppunbi  7802  gsumzaddlem  17148
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