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Theorem psrbaglesupp 16130
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 962 . . 3  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G : I --> NN0 )
2 nn0supp 10033 . . 3  |-  ( G : I --> NN0  ->  ( `' G " ( _V 
\  { 0 } ) )  =  ( `' G " NN ) )
31, 2syl 15 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  =  ( `' G " NN ) )
4 eldifi 3311 . . . . . 6  |-  ( x  e.  ( I  \ 
( `' F " NN ) )  ->  x  e.  I )
5 simpr3 963 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  G  o R  <_  F )
6 ffn 5405 . . . . . . . . . 10  |-  ( G : I --> NN0  ->  G  Fn  I )
71, 6syl 15 . . . . . . . . 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 16129 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  F  e.  D )  ->  F : I --> NN0 )
1093ad2antr1 1120 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  F : I --> NN0 )
11 ffn 5405 . . . . . . . . . 10  |-  ( F : I --> NN0  ->  F  Fn  I )
1210, 11syl 15 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  F  Fn  I )
13 simpl 443 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  I  e.  V )
14 inidm 3391 . . . . . . . . 9  |-  ( I  i^i  I )  =  I
15 eqidd 2297 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  =  ( G `  x ) )
16 eqidd 2297 . . . . . . . . 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 6102 . . . . . . . 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 201 . . . . . . 7  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  A. x  e.  I  ( G `  x )  <_  ( F `  x )
)
1918r19.21bi 2654 . . . . . 6  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  <_  ( F `  x
) )
204, 19sylan2 460 . . . . 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 10033 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
22 eqimss 3243 . . . . . . 7  |-  ( ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN )  ->  ( `' F " ( _V  \  {
0 } ) ) 
C_  ( `' F " NN ) )
2310, 21, 223syl 18 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' F " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
2410, 23suppssr 5675 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( F `  x )  =  0 )
2520, 24breqtrd 4063 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  <_  0 )
26 ffvelrn 5679 . . . . . 6  |-  ( ( G : I --> NN0  /\  x  e.  I )  ->  ( G `  x
)  e.  NN0 )
271, 4, 26syl2an 463 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  e.  NN0 )
2827nn0ge0d 10037 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  0  <_  ( G `  x
) )
2927nn0red 10035 . . . . 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 8854 . . . . 5  |-  0  e.  RR
31 letri3 8923 . . . . 5  |-  ( ( ( G `  x
)  e.  RR  /\  0  e.  RR )  ->  ( ( G `  x )  =  0  <-> 
( ( G `  x )  <_  0  /\  0  <_  ( G `
 x ) ) ) )
3229, 30, 31sylancl 643 . . . 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 888 . . 3  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  =  0 )
341, 33suppss 5674 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
353, 34eqsstr3d 3226 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Rcofr 6093    ^m cmap 6788   Fincfn 6879   RRcr 8752   0cc0 8753    <_ cle 8884   NNcn 9762   NN0cn0 9981
This theorem is referenced by:  psrbaglecl  16131  psrbagcon  16133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ofr 6095  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982
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