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Theorem suppss3 28152
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 6306 . 2  |-  ( G supp 
Z )  =  ( ( x  e.  A  |->  B ) supp  Z )
3 simpl 458 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ph )
4 eldifi 3584 . . . . 5  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
54adantl 467 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  x  e.  A
)
6 suppss3.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  A )
7 suppss3.a . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  V )
8 fnex 6138 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
96, 7, 8syl2anc 665 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  _V )
10 suppss3.z . . . . . . . . . . . . 13  |-  ( ph  ->  Z  e.  W )
11 suppimacnv 6927 . . . . . . . . . . . . 13  |-  ( ( F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
129, 10, 11syl2anc 665 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
1312eleq2d 2490 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( F supp  Z )  <->  x  e.  ( `' F " ( _V 
\  { Z }
) ) ) )
14 elpreima 6008 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
156, 14syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
1613, 15bitrd 256 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( F supp  Z )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
1716baibd 917 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ( F supp 
Z )  <->  ( F `  x )  e.  ( _V  \  { Z } ) ) )
1817notbid 295 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  x  e.  ( F supp  Z )  <->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
1918biimpd 210 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  x  e.  ( F supp  Z )  ->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
2019expimpd 606 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) )  ->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
21 eldif 3443 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  <->  ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) ) )
22 fvex 5882 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
23 eldifsn 4119 . . . . . . . 8  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
2422, 23mpbiran 926 . . . . . . 7  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
)
2524necon2bbii 2689 . . . . . 6  |-  ( ( F `  x )  =  Z  <->  -.  ( F `  x )  e.  ( _V  \  { Z } ) )
2620, 21, 253imtr4g 273 . . . . 5  |-  ( ph  ->  ( x  e.  ( A  \  ( F supp 
Z ) )  -> 
( F `  x
)  =  Z ) )
2726imp 430 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( F `  x )  =  Z )
28 suppss3.3 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  ( F `  x )  =  Z )  ->  B  =  Z )
293, 5, 27, 28syl3anc 1264 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  B  =  Z )
3029, 7suppss2 6951 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B ) supp  Z
)  C_  ( F supp  Z ) )
312, 30syl5eqss 3505 1  |-  ( ph  ->  ( G supp  Z ) 
C_  ( F supp  Z
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078    \ cdif 3430    C_ wss 3433   {csn 3993    |-> cmpt 4475   `'ccnv 4844   "cima 4848    Fn wfn 5587   ` cfv 5592  (class class class)co 6296   supp csupp 6916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-supp 6917
This theorem is referenced by:  eulerpartlems  29016
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