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Theorem suppfnss 6943
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 5686 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2527 . . . . . . . . . 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 5872 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
54eqeq1d 2459 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( G `  x
)  =  Z  <->  ( G `  y )  =  Z ) )
6 fveq2 5872 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
76eqeq1d 2459 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( F `  x
)  =  Z  <->  ( F `  y )  =  Z ) )
85, 7imbi12d 320 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  <->  ( ( G `
 y )  =  Z  ->  ( F `  y )  =  Z ) ) )
98rspcv 3206 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. x  e.  A  ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( ( G `  y )  =  Z  ->  ( F `
 y )  =  Z ) ) )
103, 9syl6bi 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 ) ) ) )
1110com23 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 ) ) ) )
1211imp31 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 ) )
1312necon3d 2681 . . . . 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 ) )
1413ss2rabdv 3577 . . . 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 } )
15 simpr1 1002 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  A  C_  B )
161ad2antrr 725 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  F  =  A )
17 fndm 5686 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
1817ad2antlr 726 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  G  =  B )
1915, 16, 183sstr4d 3542 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  F  C_  dom  G )
2019adantr 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 )
21 rabss2 3579 . . . . 5  |-  ( dom 
F  C_  dom  G  ->  { y  e.  dom  F  |  ( G `  y )  =/=  Z }  C_  { y  e. 
dom  G  |  ( G `  y )  =/=  Z } )
2220, 21syl 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 } )
2314, 22sstrd 3509 . . 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 } )
24 fnfun 5684 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
2524ad2antrr 725 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Fun  F )
26 simpl 457 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  F  Fn  A )
27 ssexg 4602 . . . . . . . 8  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
28273adant3 1016 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  A  e.  _V )
29 fnex 6140 . . . . . . 7  |-  ( ( F  Fn  A  /\  A  e.  _V )  ->  F  e.  _V )
3026, 28, 29syl2an 477 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  F  e.  _V )
31 simpr3 1004 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Z  e.  W )
32 suppval1 6923 . . . . . 6  |-  ( ( Fun  F  /\  F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  {
y  e.  dom  F  |  ( F `  y )  =/=  Z } )
3325, 30, 31, 32syl3anc 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 } )
34 fnfun 5684 . . . . . . 7  |-  ( G  Fn  B  ->  Fun  G )
3534ad2antlr 726 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Fun  G )
36 simpr 461 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  G  Fn  B )
37 simp2 997 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  B  e.  V )
38 fnex 6140 . . . . . . 7  |-  ( ( G  Fn  B  /\  B  e.  V )  ->  G  e.  _V )
3936, 37, 38syl2an 477 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  G  e.  _V )
40 suppval1 6923 . . . . . 6  |-  ( ( Fun  G  /\  G  e.  _V  /\  Z  e.  W )  ->  ( G supp  Z )  =  {
y  e.  dom  G  |  ( G `  y )  =/=  Z } )
4135, 39, 31, 40syl3anc 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 } )
4233, 41sseq12d 3528 . . . 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 } ) )
4342adantr 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 } ) )
4423, 43mpbird 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 ) )
4544ex 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   dom cdm 5008   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   supp csupp 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-supp 6918
This theorem is referenced by:  funsssuppss  6944  suppofss1d  6955  suppofss2d  6956  lincresunit2  33181
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