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Theorem suppss3 27373
Description: Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypotheses
Ref Expression
suppss3.1  |-  G  =  ( x  e.  A  |->  B )
suppss3.a  |-  ( ph  ->  A  e.  V )
suppss3.z  |-  ( ph  ->  Z  e.  W )
suppss3.2  |-  ( ph  ->  F  Fn  A )
suppss3.3  |-  ( (
ph  /\  x  e.  A  /\  ( F `  x )  =  Z )  ->  B  =  Z )
Assertion
Ref Expression
suppss3  |-  ( ph  ->  ( G supp  Z ) 
C_  ( F supp  Z
) )
Distinct variable groups:    x, A    x, F    x, Z    ph, x
Allowed substitution hints:    B( x)    G( x)    V( x)    W( x)

Proof of Theorem suppss3
StepHypRef Expression
1 suppss3.1 . . 3  |-  G  =  ( x  e.  A  |->  B )
21oveq1i 6305 . 2  |-  ( G supp 
Z )  =  ( ( x  e.  A  |->  B ) supp  Z )
3 simpl 457 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ph )
4 eldif 3491 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  <->  ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) ) )
54simplbi 460 . . . . 5  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
65adantl 466 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  x  e.  A
)
7 suppss3.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  Fn  A )
8 suppss3.a . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  V )
9 fnex 6138 . . . . . . . . . . . . . . 15  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
107, 8, 9syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
11 suppss3.z . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  W )
12 suppimacnv 6924 . . . . . . . . . . . . . 14  |-  ( ( F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
1310, 11, 12syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
1413eleq2d 2537 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( F supp  Z )  <->  x  e.  ( `' F " ( _V 
\  { Z }
) ) ) )
15 elpreima 6008 . . . . . . . . . . . . 13  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
167, 15syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
1714, 16bitrd 253 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( F supp  Z )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
1817baibd 907 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ( F supp 
Z )  <->  ( F `  x )  e.  ( _V  \  { Z } ) ) )
1918notbid 294 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  x  e.  ( F supp  Z )  <->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
2019biimpd 207 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  x  e.  ( F supp  Z )  ->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
2120expimpd 603 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) )  ->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
22 fvex 5882 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
23 eldifsn 4158 . . . . . . . . 9  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
2422, 23mpbiran 916 . . . . . . . 8  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
)
2524necon2bbii 2734 . . . . . . 7  |-  ( ( F `  x )  =  Z  <->  -.  ( F `  x )  e.  ( _V  \  { Z } ) )
2621, 25syl6ibr 227 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) )  ->  ( F `  x )  =  Z ) )
274, 26syl5bi 217 . . . . 5  |-  ( ph  ->  ( x  e.  ( A  \  ( F supp 
Z ) )  -> 
( F `  x
)  =  Z ) )
2827imp 429 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( F `  x )  =  Z )
29 suppss3.3 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  ( F `  x )  =  Z )  ->  B  =  Z )
303, 6, 28, 29syl3anc 1228 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  B  =  Z )
3130, 8suppss2 6946 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B ) supp  Z
)  C_  ( F supp  Z ) )
322, 31syl5eqss 3553 1  |-  ( ph  ->  ( G supp  Z ) 
C_  ( F supp  Z
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    \ cdif 3478    C_ wss 3481   {csn 4033    |-> cmpt 4511   `'ccnv 5004   "cima 5008    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   supp csupp 6913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-supp 6914
This theorem is referenced by:  eulerpartlems  28124
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