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Theorem suppfnss 6952
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 5693 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2492 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
y  e.  dom  F  <->  y  e.  A ) )
32ad2antrr 730 . . . . . . . . 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 5882 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
54eqeq1d 2424 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( G `  x
)  =  Z  <->  ( G `  y )  =  Z ) )
6 fveq2 5882 . . . . . . . . . . . 12  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
76eqeq1d 2424 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
( F `  x
)  =  Z  <->  ( F `  y )  =  Z ) )
85, 7imbi12d 321 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  <->  ( ( G `
 y )  =  Z  ->  ( F `  y )  =  Z ) ) )
98rspcv 3178 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. x  e.  A  ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( ( G `  y )  =  Z  ->  ( F `
 y )  =  Z ) ) )
103, 9syl6bi 231 . . . . . . . 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 81 . . . . . . 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 433 . . . . . 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 2644 . . . . 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 3542 . . . 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 1011 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  A  C_  B )
161ad2antrr 730 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  F  =  A )
17 fndm 5693 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
1817ad2antlr 731 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  G  =  B )
1915, 16, 183sstr4d 3507 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  dom  F  C_  dom  G )
2019adantr 466 . . . . 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 3544 . . . . 5  |-  ( dom 
F  C_  dom  G  ->  { y  e.  dom  F  |  ( G `  y )  =/=  Z }  C_  { y  e. 
dom  G  |  ( G `  y )  =/=  Z } )
2220, 21syl 17 . . . 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 3474 . . 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 5691 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
2524ad2antrr 730 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Fun  F )
26 simpl 458 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  F  Fn  A )
27 ssexg 4570 . . . . . . . 8  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
28273adant3 1025 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  A  e.  _V )
29 fnex 6148 . . . . . . 7  |-  ( ( F  Fn  A  /\  A  e.  _V )  ->  F  e.  _V )
3026, 28, 29syl2an 479 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  F  e.  _V )
31 simpr3 1013 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Z  e.  W )
32 suppval1 6932 . . . . . 6  |-  ( ( Fun  F  /\  F  e.  _V  /\  Z  e.  W )  ->  ( F supp  Z )  =  {
y  e.  dom  F  |  ( F `  y )  =/=  Z } )
3325, 30, 31, 32syl3anc 1264 . . . . 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 5691 . . . . . . 7  |-  ( G  Fn  B  ->  Fun  G )
3534ad2antlr 731 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  Fun  G )
36 simpr 462 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  G  Fn  B )
37 simp2 1006 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  V  /\  Z  e.  W )  ->  B  e.  V )
38 fnex 6148 . . . . . . 7  |-  ( ( G  Fn  B  /\  B  e.  V )  ->  G  e.  _V )
3936, 37, 38syl2an 479 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  G  e.  _V )
40 suppval1 6932 . . . . . 6  |-  ( ( Fun  G  /\  G  e.  _V  /\  Z  e.  W )  ->  ( G supp  Z )  =  {
y  e.  dom  G  |  ( G `  y )  =/=  Z } )
4135, 39, 31, 40syl3anc 1264 . . . . 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 3493 . . . 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 466 . . 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 235 . 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 435 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   {crab 2775   _Vcvv 3080    C_ wss 3436   dom cdm 4853   Fun wfun 5595    Fn wfn 5596   ` cfv 5601  (class class class)co 6306   supp csupp 6926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-supp 6927
This theorem is referenced by:  funsssuppss  6953  suppofss1d  6964  suppofss2d  6965  lincresunit2  39922
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