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Theorem suppfnss 6922
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 5678 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2537 . . . . . . . . . 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 5864 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
54eqeq1d 2469 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( G `  x
)  =  Z  <->  ( G `  y )  =  Z ) )
6 fveq2 5864 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
76eqeq1d 2469 . . . . . . . . . . . 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 3212 . . . . . . . . . 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 2691 . . . . 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 3581 . . . 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 1002 . . . . . . 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 5678 . . . . . . . 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 3547 . . . . . 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 3583 . . . . 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 3514 . . 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 5676 . . . . . . 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 4593 . . . . . . . 8  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
29283adant3 1016 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  A  e.  _V )
30 fnex 6125 . . . . . . 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 1004 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Z  e.  W )
33 suppval1 6904 . . . . . 6  |-  ( ( Fun  F  /\  F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  {
y  e.  dom  F  |  ( F `  y )  =/=  Z } )
3426, 31, 32, 33syl3anc 1228 . . . . 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 5676 . . . . . . 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 997 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  B  e.  V )
39 fnex 6125 . . . . . . 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 6904 . . . . . 6  |-  ( ( Fun  G  /\  G  e.  _V  /\  Z  e.  W )  ->  ( G supp  Z )  =  {
y  e.  dom  G  |  ( G `  y )  =/=  Z } )
4236, 40, 32, 41syl3anc 1228 . . . . 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 3533 . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   dom cdm 4999   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   supp csupp 6898
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 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-supp 6899
This theorem is referenced by:  funsssuppss  6923  suppofss1d  6934  suppofss2d  6935  lincresunit2  32152
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