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Theorem psrbaglesuppg 14768
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 1066 . . . . . . . . 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 14765 . . . . . . . . . 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 1067 . . . . . . . . . . 11 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G : I --> NN0 )
109ffnd 5490 . . . . . . . . . 10 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G Fn I )
11 elpreima 5775 . . . . . . . . . 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 5791 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. NN0 )
1615nn0zd 9661 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. ZZ )
17 1red 8254 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 e. RR )
18 ffun 5492 . . . . . . . . . 10 |- ( G : I --> NN0 -> Fun G )
19183ad2ant2 1046 . . . . . . . . 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 5770 . . . . . . . 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 9215 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. RR )
2415nn0red 9517 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. RR )
2522nnge1d 9245 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 <_ ( G ` x ) )
26 simplr3 1068 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G oR <_ F )
277ffnd 5490 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> F Fn I )
28 inidm 3418 . . . . . . . 8 |- ( I i^i I ) = I
29 eqidd 2232 . . . . . . . 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 2232 . . . . . . . 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 6255 . . . . . . 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 1376 . . . . . 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 8362 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 <_ ( F ` x ) )
34 elnnz1 9563 . . . . 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 5493 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> Fun F )
377fdmd 5496 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> dom F = I )
3814, 37eleqtrrd 2311 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. dom F )
39 fvimacnv 5771 . . . . 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 3234 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 1005   = wceq 1398   e. wcel 2202   {crab 2515   C_ wss 3201   class class class wbr 4093   `'ccnv 4730   dom cdm 4731   "cima 4734   Fun wfun 5327   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   oRcofr 6243   ^m cmap 6860   Fincfn 6952   1c1 8093   <_ cle 8274   NNcn 9202   NN0cn0 9461   ZZcz 9540
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-ofr 6245  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541
This theorem is referenced by:  psrbaglesupp  14769
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