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Theorem psrbaglesupp 16425
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 10265 . . 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 3461 . . . . . 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 5583 . . . . . . . . . 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 16424 . . . . . . . . . . 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 5583 . . . . . . . . . 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 3542 . . . . . . . . 9  |-  ( I  i^i  I )  =  I
15 eqidd 2436 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  =  ( G `  x ) )
16 eqidd 2436 . . . . . . . . 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 6305 . . . . . . . 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 2796 . . . . . 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 10265 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
22 eqimss 3392 . . . . . . 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 5856 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( F `  x )  =  0 )
2520, 24breqtrd 4228 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  ( G `  x )  <_  0 )
26 ffvelrn 5860 . . . . . 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 10269 . . . 4  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  /\  x  e.  ( I  \  ( `' F " NN ) ) )  ->  0  <_  ( G `  x
) )
2927nn0red 10267 . . . . 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 9083 . . . . 5  |-  0  e.  RR
31 letri3 9152 . . . . 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 5855 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F
) )  ->  ( `' G " ( _V 
\  { 0 } ) )  C_  ( `' F " NN ) )
353, 34eqsstr3d 3375 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806   class class class wbr 4204   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Rcofr 6296    ^m cmap 7010   Fincfn 7101   RRcr 8981   0cc0 8982    <_ cle 9113   NNcn 9992   NN0cn0 10213
This theorem is referenced by:  psrbaglecl  16426  psrbagcon  16428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ofr 6298  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214
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