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Theorem suppfnss 6719
Description: The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.)
Assertion
Ref Expression
suppfnss  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( A. x  e.  A  ( ( G `
 x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( F supp  Z )  C_  ( G supp  Z ) ) )
Distinct variable groups:    x, A    x, F    x, G    x, Z
Allowed substitution hints:    B( x)    V( x)    W( x)

Proof of Theorem suppfnss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fndm 5515 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2510 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
y  e.  dom  F  <->  y  e.  A ) )
32ad2antrr 725 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( y  e.  dom  F  <-> 
y  e.  A ) )
4 fveq2 5696 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
54eqeq1d 2451 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( G `  x
)  =  Z  <->  ( G `  y )  =  Z ) )
6 fveq2 5696 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
76eqeq1d 2451 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( F `  x
)  =  Z  <->  ( F `  y )  =  Z ) )
85, 7imbi12d 320 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  <->  ( ( G `
 y )  =  Z  ->  ( F `  y )  =  Z ) ) )
98rspcva 3076 . . . . . . . . . 10  |-  ( ( y  e.  A  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  ( ( G `  y )  =  Z  ->  ( F `
 y )  =  Z ) )
109ex 434 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. x  e.  A  ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( ( G `  y )  =  Z  ->  ( F `
 y )  =  Z ) ) )
113, 10syl6bi 228 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( y  e.  dom  F  ->  ( A. x  e.  A  ( ( G `  x )  =  Z  ->  ( F `
 x )  =  Z )  ->  (
( G `  y
)  =  Z  -> 
( F `  y
)  =  Z ) ) ) )
1211com23 78 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( A. x  e.  A  ( ( G `
 x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( y  e.  dom  F  ->  (
( G `  y
)  =  Z  -> 
( F `  y
)  =  Z ) ) ) )
1312imp31 432 . . . . . 6  |-  ( ( ( ( ( F  Fn  A  /\  G  Fn  B )  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  /\  y  e. 
dom  F )  -> 
( ( G `  y )  =  Z  ->  ( F `  y )  =  Z ) )
1413necon3d 2651 . . . . 5  |-  ( ( ( ( ( F  Fn  A  /\  G  Fn  B )  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  /\  y  e. 
dom  F )  -> 
( ( F `  y )  =/=  Z  ->  ( G `  y
)  =/=  Z ) )
1514ss2rabdv 3438 . . . 4  |-  ( ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  { y  e.  dom  F  |  ( F `  y )  =/=  Z }  C_  { y  e.  dom  F  |  ( G `  y )  =/=  Z } )
16 simpr1 994 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  A  C_  B )
171ad2antrr 725 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  F  =  A )
18 fndm 5515 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
1918ad2antlr 726 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  G  =  B )
2016, 17, 193sstr4d 3404 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  F  C_  dom  G )
2120adantr 465 . . . . 5  |-  ( ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  dom  F  C_  dom  G )
22 rabss2 3440 . . . . 5  |-  ( dom 
F  C_  dom  G  ->  { y  e.  dom  F  |  ( G `  y )  =/=  Z }  C_  { y  e. 
dom  G  |  ( G `  y )  =/=  Z } )
2321, 22syl 16 . . . 4  |-  ( ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  { y  e.  dom  F  |  ( G `  y )  =/=  Z }  C_  { y  e.  dom  G  |  ( G `  y )  =/=  Z } )
2415, 23sstrd 3371 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  { y  e.  dom  F  |  ( F `  y )  =/=  Z }  C_  { y  e.  dom  G  |  ( G `  y )  =/=  Z } )
25 fnfun 5513 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
2625ad2antrr 725 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Fun  F )
27 simpl 457 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  F  Fn  A )
28 ssexg 4443 . . . . . . . 8  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
29283adant3 1008 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  A  e.  _V )
30 fnex 5949 . . . . . . 7  |-  ( ( F  Fn  A  /\  A  e.  _V )  ->  F  e.  _V )
3127, 29, 30syl2an 477 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  F  e.  _V )
32 simpr3 996 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Z  e.  W )
33 suppval1 6701 . . . . . 6  |-  ( ( Fun  F  /\  F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  {
y  e.  dom  F  |  ( F `  y )  =/=  Z } )
3426, 31, 32, 33syl3anc 1218 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( F supp  Z )  =  { y  e.  dom  F  |  ( F `  y )  =/=  Z } )
35 fnfun 5513 . . . . . . 7  |-  ( G  Fn  B  ->  Fun  G )
3635ad2antlr 726 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Fun  G )
37 simpr 461 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  G  Fn  B )
38 simp2 989 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  B  e.  V )
39 fnex 5949 . . . . . . 7  |-  ( ( G  Fn  B  /\  B  e.  V )  ->  G  e.  _V )
4037, 38, 39syl2an 477 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  G  e.  _V )
41 suppval1 6701 . . . . . 6  |-  ( ( Fun  G  /\  G  e.  _V  /\  Z  e.  W )  ->  ( G supp  Z )  =  {
y  e.  dom  G  |  ( G `  y )  =/=  Z } )
4236, 40, 32, 41syl3anc 1218 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( G supp  Z )  =  { y  e.  dom  G  |  ( G `  y )  =/=  Z } )
4334, 42sseq12d 3390 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( ( F supp  Z
)  C_  ( G supp  Z )  <->  { y  e.  dom  F  |  ( F `  y )  =/=  Z }  C_  { y  e. 
dom  G  |  ( G `  y )  =/=  Z } ) )
4443adantr 465 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  ( ( F supp  Z )  C_  ( G supp  Z )  <->  { y  e.  dom  F  |  ( F `  y )  =/=  Z }  C_  { y  e.  dom  G  |  ( G `  y )  =/=  Z } ) )
4524, 44mpbird 232 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  /\  A. x  e.  A  ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )  ->  ( F supp  Z )  C_  ( G supp  Z ) )
4645ex 434 1  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  -> 
( A. x  e.  A  ( ( G `
 x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( F supp  Z )  C_  ( G supp  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977    C_ wss 3333   dom cdm 4845   Fun wfun 5417    Fn wfn 5418   ` cfv 5423  (class class class)co 6096   supp csupp 6695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-supp 6696
This theorem is referenced by:  funsssuppss  6720  suppofss1d  6731  suppofss2d  6732  lincresunit2  31017
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