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Theorem psrbaglesupp 16435
Description: The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
psrbag.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
Assertion
Ref Expression
psrbaglesupp  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " NN ) 
C_  ( `' F " NN ) )
Distinct variable groups:    f, F    f, G    f, I
Allowed substitution hints:    D( f)    V( f)

Proof of Theorem psrbaglesupp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr2 965 . . 3  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G : I --> NN0 )
2 nn0supp 10275 . . 3  |-  ( G : I --> NN0  ->  ( `' G " ( _V 
\  { 0 } ) )  =  ( `' G " NN ) )
31, 2syl 16 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  =  ( `' G " NN ) )
4 eldifi 3471 . . . . . 6  |-  ( x  e.  ( I  \ 
( `' F " NN ) )  ->  x  e.  I )
5 simpr3 966 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G  o R  <_  F )
6 ffn 5593 . . . . . . . . . 10  |-  ( G : I --> NN0  ->  G  Fn  I )
71, 6syl 16 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G  Fn  I )
8 psrbag.d . . . . . . . . . . . 12  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
98psrbagf 16434 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  F  e.  D )  ->  F : I --> NN0 )
1093ad2antr1 1123 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  F : I --> NN0 )
11 ffn 5593 . . . . . . . . . 10  |-  ( F : I --> NN0  ->  F  Fn  I )
1210, 11syl 16 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  F  Fn  I )
13 simpl 445 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  I  e.  V )
14 inidm 3552 . . . . . . . . 9  |-  ( I  i^i  I )  =  I
15 eqidd 2439 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  =  ( G `  x ) )
16 eqidd 2439 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( F `  x )  =  ( F `  x ) )
177, 12, 13, 13, 14, 15, 16ofrfval 6315 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( G  o R  <_  F  <->  A. x  e.  I  ( G `  x )  <_  ( F `  x ) ) )
185, 17mpbid 203 . . . . . . 7  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  A. x  e.  I  ( G `  x )  <_  ( F `  x )
)
1918r19.21bi 2806 . . . . . 6  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  <_  ( F `  x
) )
204, 19sylan2 462 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  <_  ( F `  x
) )
21 nn0supp 10275 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
22 eqimss 3402 . . . . . . 7  |-  ( ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN )  ->  ( `' F " ( _V  \  {
0 } ) ) 
C_  ( `' F " NN ) )
2310, 21, 223syl 19 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' F " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
2410, 23suppssr 5866 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( F `  x )  =  0 )
2520, 24breqtrd 4238 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  <_  0 )
26 ffvelrn 5870 . . . . . 6  |-  ( ( G : I --> NN0  /\  x  e.  I )  ->  ( G `  x
)  e.  NN0 )
271, 4, 26syl2an 465 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  e.  NN0 )
2827nn0ge0d 10279 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  0  <_  ( G `  x
) )
2927nn0red 10277 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  e.  RR )
30 0re 9093 . . . . 5  |-  0  e.  RR
31 letri3 9162 . . . . 5  |-  ( ( ( G `  x
)  e.  RR  /\  0  e.  RR )  ->  ( ( G `  x )  =  0  <-> 
( ( G `  x )  <_  0  /\  0  <_  ( G `
 x ) ) ) )
3229, 30, 31sylancl 645 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  (
( G `  x
)  =  0  <->  (
( G `  x
)  <_  0  /\  0  <_  ( G `  x ) ) ) )
3325, 28, 32mpbir2and 890 . . 3  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  =  0 )
341, 33suppss 5865 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
353, 34eqsstr3d 3385 1  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " NN ) 
C_  ( `' F " NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816   class class class wbr 4214   `'ccnv 4879   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Rcofr 6306    ^m cmap 7020   Fincfn 7111   RRcr 8991   0cc0 8992    <_ cle 9123   NNcn 10002   NN0cn0 10223
This theorem is referenced by:  psrbaglecl  16436  psrbagcon  16438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-ofr 6308  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224
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