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Theorem psrbaglesupp 16360
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 964 . . 3  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G : I --> NN0 )
2 nn0supp 10205 . . 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 3412 . . . . . 6  |-  ( x  e.  ( I  \ 
( `' F " NN ) )  ->  x  e.  I )
5 simpr3 965 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G  o R  <_  F )
6 ffn 5531 . . . . . . . . . 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 16359 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  F  e.  D )  ->  F : I --> NN0 )
1093ad2antr1 1122 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  F : I --> NN0 )
11 ffn 5531 . . . . . . . . . 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 444 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  I  e.  V )
14 inidm 3493 . . . . . . . . 9  |-  ( I  i^i  I )  =  I
15 eqidd 2388 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  =  ( G `  x ) )
16 eqidd 2388 . . . . . . . . 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 6252 . . . . . . . 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 202 . . . . . . 7  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  A. x  e.  I  ( G `  x )  <_  ( F `  x )
)
1918r19.21bi 2747 . . . . . 6  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  <_  ( F `  x
) )
204, 19sylan2 461 . . . . 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 10205 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
22 eqimss 3343 . . . . . . 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 5803 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( F `  x )  =  0 )
2520, 24breqtrd 4177 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  <_  0 )
26 ffvelrn 5807 . . . . . 6  |-  ( ( G : I --> NN0  /\  x  e.  I )  ->  ( G `  x
)  e.  NN0 )
271, 4, 26syl2an 464 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  e.  NN0 )
2827nn0ge0d 10209 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  0  <_  ( G `  x
) )
2927nn0red 10207 . . . . 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 9024 . . . . 5  |-  0  e.  RR
31 letri3 9093 . . . . 5  |-  ( ( ( G `  x
)  e.  RR  /\  0  e.  RR )  ->  ( ( G `  x )  =  0  <-> 
( ( G `  x )  <_  0  /\  0  <_  ( G `
 x ) ) ) )
3229, 30, 31sylancl 644 . . . 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 889 . . 3  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  =  0 )
341, 33suppss 5802 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
353, 34eqsstr3d 3326 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899    \ cdif 3260    C_ wss 3263   {csn 3757   class class class wbr 4153   `'ccnv 4817   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Rcofr 6243    ^m cmap 6954   Fincfn 7045   RRcr 8922   0cc0 8923    <_ cle 9054   NNcn 9932   NN0cn0 10153
This theorem is referenced by:  psrbaglecl  16361  psrbagcon  16363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-ofr 6245  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154
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