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Theorem suppss3 28312
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 6300 . 2  |-  ( G supp 
Z )  =  ( ( x  e.  A  |->  B ) supp  Z )
3 simpl 459 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ph )
4 eldifi 3555 . . . . 5  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
54adantl 468 . . . 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 6132 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
96, 7, 8syl2anc 667 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  _V )
10 suppss3.z . . . . . . . . . . . . 13  |-  ( ph  ->  Z  e.  W )
11 suppimacnv 6925 . . . . . . . . . . . . 13  |-  ( ( F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
129, 10, 11syl2anc 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  Z )  =  ( `' F " ( _V  \  { Z } ) ) )
1312eleq2d 2514 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( F supp  Z )  <->  x  e.  ( `' F " ( _V 
\  { Z }
) ) ) )
14 elpreima 6002 . . . . . . . . . . . 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 257 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( F supp  Z )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
1716baibd 920 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ( F supp 
Z )  <->  ( F `  x )  e.  ( _V  \  { Z } ) ) )
1817notbid 296 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  x  e.  ( F supp  Z )  <->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
1918biimpd 211 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( -.  x  e.  ( F supp  Z )  ->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
2019expimpd 608 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) )  ->  -.  ( F `  x )  e.  ( _V  \  { Z } ) ) )
21 eldif 3414 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  <->  ( x  e.  A  /\  -.  x  e.  ( F supp  Z ) ) )
22 fvex 5875 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
23 eldifsn 4097 . . . . . . . 8  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
2422, 23mpbiran 929 . . . . . . 7  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
)
2524necon2bbii 2675 . . . . . 6  |-  ( ( F `  x )  =  Z  <->  -.  ( F `  x )  e.  ( _V  \  { Z } ) )
2620, 21, 253imtr4g 274 . . . . 5  |-  ( ph  ->  ( x  e.  ( A  \  ( F supp 
Z ) )  -> 
( F `  x
)  =  Z ) )
2726imp 431 . . . 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 1268 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  B  =  Z )
3029, 7suppss2 6949 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B ) supp  Z
)  C_  ( F supp  Z ) )
312, 30syl5eqss 3476 1  |-  ( ph  ->  ( G supp  Z ) 
C_  ( F supp  Z
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045    \ cdif 3401    C_ wss 3404   {csn 3968    |-> cmpt 4461   `'ccnv 4833   "cima 4837    Fn wfn 5577   ` cfv 5582  (class class class)co 6290   supp csupp 6914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-supp 6915
This theorem is referenced by:  eulerpartlems  29193
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