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Theorem psrbaglesupp 16388
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 10229 . . 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 3429 . . . . . 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 5550 . . . . . . . . . 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 16387 . . . . . . . . . . 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 5550 . . . . . . . . . 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 3510 . . . . . . . . 9  |-  ( I  i^i  I )  =  I
15 eqidd 2405 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  =  ( G `  x ) )
16 eqidd 2405 . . . . . . . . 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 6272 . . . . . . . 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 2764 . . . . . 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 10229 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
22 eqimss 3360 . . . . . . 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 5823 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( F `  x )  =  0 )
2520, 24breqtrd 4196 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  <_  0 )
26 ffvelrn 5827 . . . . . 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 10233 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  0  <_  ( G `  x
) )
2927nn0red 10231 . . . . 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 9047 . . . . 5  |-  0  e.  RR
31 letri3 9116 . . . . 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 5822 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
353, 34eqsstr3d 3343 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 1721   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   class class class wbr 4172   `'ccnv 4836   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Rcofr 6263    ^m cmap 6977   Fincfn 7068   RRcr 8945   0cc0 8946    <_ cle 9077   NNcn 9956   NN0cn0 10177
This theorem is referenced by:  psrbaglecl  16389  psrbagcon  16391
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-ofr 6265  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178
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