Theorem psrbaglesuppg 14549

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.

Step	Hyp	Ref	Expression
1		simplr1 1042	. . . . . . . . 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 }
4	3	psrbag 14546	. . . . . . . . . 10 |- ( I e. V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
5	2, 4	syl 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 ) ) )
6	1, 5	mpbid 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 ) )
7	6	simpld 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 1043	. . . . . . . . . . 11 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G : I --> NN0 )
10	9	ffnd 5446	. . . . . . . . . 10 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G Fn I )
11		elpreima 5722	. . . . . . . . . 10 |- ( G Fn I -> ( x e. ( `' G " NN ) <-> ( x e. I /\ ( G ` x ) e. NN ) ) )
12	10, 11	syl 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 ) ) )
13	8, 12	mpbid 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 ) )
14	13	simpld 112	. . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. I )
15	7, 14	ffvelcdmd 5739	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. NN0 )
16	15	nn0zd 9528	. . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. ZZ )
17		1red 8122	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 e. RR )
18		ffun 5448	. . . . . . . . . 10 |- ( G : I --> NN0 -> Fun G )
19	18	3ad2ant2 1022	. . . . . . . . 9 |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> Fun G )
20	19	ad2antlr 489	. . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> Fun G )
21		fvimacnvi 5717	. . . . . . . 8 |- ( ( Fun G /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. NN )
22	20, 8, 21	syl2anc 411	. . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. NN )
23	22	nnred 9084	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) e. RR )
24	15	nn0red 9384	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. RR )
25	22	nnge1d 9114	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 <_ ( G ` x ) )
26		simplr3 1044	. . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> G oR <_ F )
27	7	ffnd 5446	. . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> F Fn I )
28		inidm 3390	. . . . . . . 8 |- ( I i^i I ) = I
29		eqidd 2208	. . . . . . . 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 2208	. . . . . . . 8 |- ( ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) /\ x e. I ) -> ( F ` x ) = ( F ` x ) )
31	10, 27, 2, 2, 28, 29, 30	ofrval 6192	. . . . . . 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 ) )
32	26, 14, 31	mpd3an23 1352	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( G ` x ) <_ ( F ` x ) )
33	17, 23, 24, 25, 32	letrd 8231	. . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> 1 <_ ( F ` x ) )
34		elnnz1 9430	. . . . 5 |- ( ( F ` x ) e. NN <-> ( ( F ` x ) e. ZZ /\ 1 <_ ( F ` x ) ) )
35	16, 33, 34	sylanbrc 417	. . . 4 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> ( F ` x ) e. NN )
36	7	ffund 5449	. . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> Fun F )
37	7	fdmd 5452	. . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> dom F = I )
38	14, 37	eleqtrrd 2287	. . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. dom F )
39		fvimacnv 5718	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. NN <-> x e. ( `' F " NN ) ) )
40	36, 38, 39	syl2anc 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 ) ) )
41	35, 40	mpbid 147	. . 3 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. ( `' G " NN ) ) -> x e. ( `' F " NN ) )
42	41	ex 115	. 2 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( x e. ( `' G " NN ) -> x e. ( `' F " NN ) ) )
43	42	ssrdv 3207	1 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' G " NN ) C_ ( `' F " NN ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   {crab 2490   C_ wss 3174   class class class wbr 4059   `'ccnv 4692   dom cdm 4693   "cima 4696   Fun wfun 5284   Fn wfn 5285   -->wf 5286   ` cfv 5290  (class class class)co 5967   oRcofr 6180   ^m cmap 6758   Fincfn 6850   1c1 7961   <_ cle 8143   NNcn 9071   NN0cn0 9330   ZZcz 9407

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-ofr 6182  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408

This theorem is referenced by: (None)