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Theorem funsssuppss 6826
Description: The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
Assertion
Ref Expression
funsssuppss  |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  ( F supp  Z )  C_  ( G supp  Z ) )

Proof of Theorem funsssuppss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funss 5545 . . . . . . . . . 10  |-  ( F 
C_  G  ->  ( Fun  G  ->  Fun  F ) )
21impcom 430 . . . . . . . . 9  |-  ( ( Fun  G  /\  F  C_  G )  ->  Fun  F )
3 funfn 5556 . . . . . . . . . 10  |-  ( Fun 
F  <->  F  Fn  dom  F )
43biimpi 194 . . . . . . . . 9  |-  ( Fun 
F  ->  F  Fn  dom  F )
52, 4syl 16 . . . . . . . 8  |-  ( ( Fun  G  /\  F  C_  G )  ->  F  Fn  dom  F )
6 funfn 5556 . . . . . . . . . 10  |-  ( Fun 
G  <->  G  Fn  dom  G )
76biimpi 194 . . . . . . . . 9  |-  ( Fun 
G  ->  G  Fn  dom  G )
87adantr 465 . . . . . . . 8  |-  ( ( Fun  G  /\  F  C_  G )  ->  G  Fn  dom  G )
95, 8jca 532 . . . . . . 7  |-  ( ( Fun  G  /\  F  C_  G )  ->  ( F  Fn  dom  F  /\  G  Fn  dom  G ) )
1093adant3 1008 . . . . . 6  |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  ( F  Fn  dom  F  /\  G  Fn  dom  G ) )
1110adantr 465 . . . . 5  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  ( F  Fn  dom  F  /\  G  Fn  dom  G ) )
12 dmss 5148 . . . . . . . 8  |-  ( F 
C_  G  ->  dom  F 
C_  dom  G )
13123ad2ant2 1010 . . . . . . 7  |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  dom  F 
C_  dom  G )
1413adantr 465 . . . . . 6  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  dom  F 
C_  dom  G )
15 dmexg 6620 . . . . . . . 8  |-  ( G  e.  V  ->  dom  G  e.  _V )
16153ad2ant3 1011 . . . . . . 7  |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  dom  G  e.  _V )
1716adantr 465 . . . . . 6  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  dom  G  e.  _V )
18 simpr 461 . . . . . 6  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  Z  e.  _V )
1914, 17, 183jca 1168 . . . . 5  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  ( dom  F  C_  dom  G  /\  dom  G  e.  _V  /\  Z  e.  _V )
)
2011, 19jca 532 . . . 4  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  (
( F  Fn  dom  F  /\  G  Fn  dom  G )  /\  ( dom 
F  C_  dom  G  /\  dom  G  e.  _V  /\  Z  e.  _V )
) )
21 funssfv 5815 . . . . . . . . 9  |-  ( ( Fun  G  /\  F  C_  G  /\  x  e. 
dom  F )  -> 
( G `  x
)  =  ( F `
 x ) )
22213expa 1188 . . . . . . . 8  |-  ( ( ( Fun  G  /\  F  C_  G )  /\  x  e.  dom  F )  ->  ( G `  x )  =  ( F `  x ) )
23 eqeq1 2458 . . . . . . . . 9  |-  ( ( G `  x )  =  ( F `  x )  ->  (
( G `  x
)  =  Z  <->  ( F `  x )  =  Z ) )
2423biimpd 207 . . . . . . . 8  |-  ( ( G `  x )  =  ( F `  x )  ->  (
( G `  x
)  =  Z  -> 
( F `  x
)  =  Z ) )
2522, 24syl 16 . . . . . . 7  |-  ( ( ( Fun  G  /\  F  C_  G )  /\  x  e.  dom  F )  ->  ( ( G `
 x )  =  Z  ->  ( F `  x )  =  Z ) )
2625ralrimiva 2830 . . . . . 6  |-  ( ( Fun  G  /\  F  C_  G )  ->  A. x  e.  dom  F ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z ) )
27263adant3 1008 . . . . 5  |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  A. x  e.  dom  F ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z ) )
2827adantr 465 . . . 4  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  A. x  e.  dom  F ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z ) )
29 suppfnss 6825 . . . 4  |-  ( ( ( F  Fn  dom  F  /\  G  Fn  dom  G )  /\  ( dom 
F  C_  dom  G  /\  dom  G  e.  _V  /\  Z  e.  _V )
)  ->  ( A. x  e.  dom  F ( ( G `  x
)  =  Z  -> 
( F `  x
)  =  Z )  ->  ( F supp  Z
)  C_  ( G supp  Z ) ) )
3020, 28, 29sylc 60 . . 3  |-  ( ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  /\  Z  e.  _V )  ->  ( F supp  Z )  C_  ( G supp  Z ) )
3130expcom 435 . 2  |-  ( Z  e.  _V  ->  (
( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  ( F supp  Z )  C_  ( G supp  Z ) ) )
32 ssid 3484 . . . 4  |-  (/)  C_  (/)
33 simpr 461 . . . . . . 7  |-  ( ( F  e.  _V  /\  Z  e.  _V )  ->  Z  e.  _V )
3433con3i 135 . . . . . 6  |-  ( -.  Z  e.  _V  ->  -.  ( F  e.  _V  /\  Z  e.  _V )
)
35 supp0prc 6804 . . . . . 6  |-  ( -.  ( F  e.  _V  /\  Z  e.  _V )  ->  ( F supp  Z )  =  (/) )
3634, 35syl 16 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F supp  Z )  =  (/) )
37 simpr 461 . . . . . . 7  |-  ( ( G  e.  _V  /\  Z  e.  _V )  ->  Z  e.  _V )
3837con3i 135 . . . . . 6  |-  ( -.  Z  e.  _V  ->  -.  ( G  e.  _V  /\  Z  e.  _V )
)
39 supp0prc 6804 . . . . . 6  |-  ( -.  ( G  e.  _V  /\  Z  e.  _V )  ->  ( G supp  Z )  =  (/) )
4038, 39syl 16 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( G supp  Z )  =  (/) )
4136, 40sseq12d 3494 . . . 4  |-  ( -.  Z  e.  _V  ->  ( ( F supp  Z ) 
C_  ( G supp  Z
)  <->  (/)  C_  (/) ) )
4232, 41mpbiri 233 . . 3  |-  ( -.  Z  e.  _V  ->  ( F supp  Z )  C_  ( G supp  Z )
)
4342a1d 25 . 2  |-  ( -.  Z  e.  _V  ->  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  ( F supp  Z )  C_  ( G supp  Z ) ) )
4431, 43pm2.61i 164 1  |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  ( F supp  Z )  C_  ( G supp  Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    C_ wss 3437   (/)c0 3746   dom cdm 4949   Fun wfun 5521    Fn wfn 5522   ` cfv 5527  (class class class)co 6201   supp csupp 6801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-supp 6802
This theorem is referenced by:  tdeglem4  21663
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