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Theorem psrbaglesuppg 14158
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
psrbaglesuppg |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' G " NN ) C_ ( `' F " NN ) )
Distinct variable groups:   f, F   f, I
Allowed substitution hints:   D( f)   G( f)   V( f)
Proof of Theorem psrbaglesuppg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simplr1 1041 . . . . . . . . 9 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> F e. D )
2 simpll 527 . . . . . . . . . 10 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> I e. V )
3 psrbag.d . . . . . . . . . . 11 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
43psrbag 14155 . . . . . . . . . 10 |- ( I e. V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
52, 4syl 14 . . . . . . . . 9 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
61, 5mpbid 147 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) )
76simpld 112 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> F : I --> NN0 )
8 simpr 110 . . . . . . . . 9 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. ( `' G " NN ) )
9 simplr2 1042 . . . . . . . . . . 11 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G : I --> NN0 )
109ffnd 5404 . . . . . . . . . 10 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G Fn I )
11 elpreima 5677 . . . . . . . . . 10 |- ( G Fn I -> ( x e. ( `' G " NN ) <-> ( x e. I /\ ( G ` x ) e. NN ) ) )
1210, 11syl 14 . . . . . . . . 9 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( x e. ( `' G " NN ) <-> ( x e. I /\ ( G ` x ) e. NN ) ) )
138, 12mpbid 147 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( x e. I /\ ( G ` x ) e. NN ) )
1413simpld 112 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. I )
157, 14ffvelcdmd 5694 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. NN0 )
1615nn0zd 9437 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. ZZ )
17 1red 8034 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 e. RR )
18 ffun 5406 . . . . . . . . . 10 |- ( G : I --> NN0 -> Fun G )
19183ad2ant2 1021 . . . . . . . . 9 |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> Fun G )
2019ad2antlr 489 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> Fun G )
21 fvimacnvi 5672 . . . . . . . 8 |- ( ( Fun G /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. NN )
2220, 8, 21syl2anc 411 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. NN )
2322nnred 8995 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. RR )
2415nn0red 9294 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. RR )
2522nnge1d 9025 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 <_ ( G ` x ) )
26 simplr3 1043 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G oR <_ F )
277ffnd 5404 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> F Fn I )
28 inidm 3368 . . . . . . . 8 |- ( I i^i I ) = I
29 eqidd 2194 . . . . . . . 8 |- ( ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) /\ x e. I ) -> ( G ` x ) = ( G ` x ) )
30 eqidd 2194 . . . . . . . 8 |- ( ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) /\ x e. I ) -> ( F ` x ) = ( F ` x ) )
3110, 27, 2, 2, 28, 29, 30ofrval 6141 . . . . . . 7 |- ( ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) /\ G oR <_ F /\ x e. I ) -> ( G ` x ) <_ ( F ` x ) )
3226, 14, 31mpd3an23 1350 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) <_ ( F ` x ) )
3317, 23, 24, 25, 32letrd 8143 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 <_ ( F ` x ) )
34 elnnz1 9340 . . . . 5 |- ( ( F ` x ) e. NN <-> ( ( F ` x ) e. ZZ /\ 1 <_ ( F ` x ) ) )
3516, 33, 34sylanbrc 417 . . . 4 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. NN )
367ffund 5407 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> Fun F )
377fdmd 5410 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> dom F = I )
3814, 37eleqtrrd 2273 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. dom F )
39 fvimacnv 5673 . . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. NN <-> x e. ( `' F " NN ) ) )
4036, 38, 39syl2anc 411 . . . 4 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( ( F ` x ) e. NN <-> x e. ( `' F " NN ) ) )
4135, 40mpbid 147 . . 3 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. ( `' F " NN ) )
4241ex 115 . 2 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( x e. ( `' G " NN ) -> x e. ( `' F " NN ) ) )
4342ssrdv 3185 1 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' G " NN ) C_ ( `' F " NN ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {crab 2476   C_ wss 3153   class class class wbr 4029   `'ccnv 4658   dom cdm 4659   "cima 4662   Fun wfun 5248   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   oRcofr 6129   ^m cmap 6702   Fincfn 6794   1c1 7873   <_ cle 8055   NNcn 8982   NN0cn0 9240   ZZcz 9317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-ofr 6131  df-map 6704  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318
This theorem is referenced by: (None)
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