Theorem psrbagconf1o 14990

Description: Bag complementation is a bijection on the set of bags dominated by a given bag F. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)

Hypotheses

Ref	Expression
psrbag.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
psrbagconf1o.s	|- S = { y e. D | y oR <_ F }

Assertion

Ref	Expression
psrbagconf1o	|- ( F e. D -> ( x e. S |-> ( F oF - x ) ) : S -1-1-onto-> S )

Distinct variable groups:   f, F   f, I   x, D, y   x, F, y   x, I, f   x, S

Allowed substitution hints:   D( f)   S( y, f)   I( y)

Proof of Theorem psrbagconf1o

Dummy variables n z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. 2 |- ( x e. S |-> ( F oF - x ) ) = ( x e. S |-> ( F oF - x ) )
2		psrbag.d	. . 3 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
3		psrbagconf1o.s	. . 3 |- S = { y e. D | y oR <_ F }
4	2, 3	psrbagconcl 14989	. 2 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) e. S )
5	2, 3	psrbagconcl 14989	. 2 |- ( ( F e. D /\ z e. S ) -> ( F oF - z ) e. S )
6	2	psrbagf 14980	. . . . . . . . 9 |- ( F e. D -> F : I --> NN0 )
7	6	adantr 276	. . . . . . . 8 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> F : I --> NN0 )
8	7	ffvelcdmda 5837	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) e. NN0 )
9	3	ssrab3 3334	. . . . . . . . . . . 12 |- S C_ D
10	9	sseli 3244	. . . . . . . . . . 11 |- ( z e. S -> z e. D )
11	10	adantl 277	. . . . . . . . . 10 |- ( ( F e. D /\ z e. S ) -> z e. D )
12	2	psrbagf 14980	. . . . . . . . . 10 |- ( z e. D -> z : I --> NN0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( F e. D /\ z e. S ) -> z : I --> NN0 )
14	13	adantrl 482	. . . . . . . 8 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> z : I --> NN0 )
15	14	ffvelcdmda 5837	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) e. NN0 )
16		simprl 535	. . . . . . . . . 10 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x e. S )
17	9, 16	sselid 3246	. . . . . . . . 9 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x e. D )
18	2	psrbagf 14980	. . . . . . . . 9 |- ( x e. D -> x : I --> NN0 )
19	17, 18	syl 14	. . . . . . . 8 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x : I --> NN0 )
20	19	ffvelcdmda 5837	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) e. NN0 )
21		nn0cn 9555	. . . . . . . 8 |- ( ( F ` n ) e. NN0 -> ( F ` n ) e. CC )
22		nn0cn 9555	. . . . . . . 8 |- ( ( z ` n ) e. NN0 -> ( z ` n ) e. CC )
23		nn0cn 9555	. . . . . . . 8 |- ( ( x ` n ) e. NN0 -> ( x ` n ) e. CC )
24		subsub23 8524	. . . . . . . 8 |- ( ( ( F ` n ) e. CC /\ ( z ` n ) e. CC /\ ( x ` n ) e. CC ) -> ( ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) )
25	21, 22, 23, 24	syl3an 1320	. . . . . . 7 |- ( ( ( F ` n ) e. NN0 /\ ( z ` n ) e. NN0 /\ ( x ` n ) e. NN0 ) -> ( ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) )
26	8, 15, 20, 25	syl3anc 1278	. . . . . 6 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) )
27		eqcom 2240	. . . . . 6 |- ( ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) <-> ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) )
28		eqcom 2240	. . . . . 6 |- ( ( z ` n ) = ( ( F ` n ) - ( x ` n ) ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) )
29	26, 27, 28	3bitr4g 223	. . . . 5 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) <-> ( z ` n ) = ( ( F ` n ) - ( x ` n ) ) ) )
30	6	ffnd 5532	. . . . . . . 8 |- ( F e. D -> F Fn I )
31	30	adantr 276	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> F Fn I )
32	13	ffnd 5532	. . . . . . . 8 |- ( ( F e. D /\ z e. S ) -> z Fn I )
33	32	adantrl 482	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> z Fn I )
34	19	ffnd 5532	. . . . . . . 8 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x Fn I )
35	16, 34	fndmexd 5579	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> I e. _V )
36		inidm 3440	. . . . . . 7 |- ( I i^i I ) = I
37		eqidd 2239	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) = ( F ` n ) )
38		eqidd 2239	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) = ( z ` n ) )
39	8	nn0zd 9748	. . . . . . . 8 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) e. ZZ )
40	15	nn0zd 9748	. . . . . . . 8 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) e. ZZ )
41	39, 40	zsubcld 9755	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F ` n ) - ( z ` n ) ) e. ZZ )
42	31, 33, 35, 35, 36, 37, 38, 41	ofvalg 6305	. . . . . 6 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F oF - z ) ` n ) = ( ( F ` n ) - ( z ` n ) ) )
43	42	eqeq2d 2250	. . . . 5 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( x ` n ) = ( ( F oF - z ) ` n ) <-> ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) ) )
44		eqidd 2239	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) = ( x ` n ) )
45	20	nn0zd 9748	. . . . . . . 8 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) e. ZZ )
46	39, 45	zsubcld 9755	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F ` n ) - ( x ` n ) ) e. ZZ )
47	31, 34, 35, 35, 36, 37, 44, 46	ofvalg 6305	. . . . . 6 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F oF - x ) ` n ) = ( ( F ` n ) - ( x ` n ) ) )
48	47	eqeq2d 2250	. . . . 5 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( z ` n ) = ( ( F oF - x ) ` n ) <-> ( z ` n ) = ( ( F ` n ) - ( x ` n ) ) ) )
49	29, 43, 48	3bitr4d 220	. . . 4 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( x ` n ) = ( ( F oF - z ) ` n ) <-> ( z ` n ) = ( ( F oF - x ) ` n ) ) )
50	49	ralbidva 2546	. . 3 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( A. n e. I ( x ` n ) = ( ( F oF - z ) ` n ) <-> A. n e. I ( z ` n ) = ( ( F oF - x ) ` n ) ) )
51	5	adantrl 482	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) e. S )
52	9, 51	sselid 3246	. . . . . 6 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) e. D )
53	2	psrbagf 14980	. . . . . 6 |- ( ( F oF - z ) e. D -> ( F oF - z ) : I --> NN0 )
54	52, 53	syl 14	. . . . 5 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) : I --> NN0 )
55	54	ffnd 5532	. . . 4 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) Fn I )
56		eqfnfv 5800	. . . 4 |- ( ( x Fn I /\ ( F oF - z ) Fn I ) -> ( x = ( F oF - z ) <-> A. n e. I ( x ` n ) = ( ( F oF - z ) ` n ) ) )
57	34, 55, 56	syl2anc 415	. . 3 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( x = ( F oF - z ) <-> A. n e. I ( x ` n ) = ( ( F oF - z ) ` n ) ) )
58	9, 4	sselid 3246	. . . . . . 7 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) e. D )
59	2	psrbagf 14980	. . . . . . 7 |- ( ( F oF - x ) e. D -> ( F oF - x ) : I --> NN0 )
60	58, 59	syl 14	. . . . . 6 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) : I --> NN0 )
61	60	ffnd 5532	. . . . 5 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) Fn I )
62	61	adantrr 483	. . . 4 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - x ) Fn I )
63		eqfnfv 5800	. . . 4 |- ( ( z Fn I /\ ( F oF - x ) Fn I ) -> ( z = ( F oF - x ) <-> A. n e. I ( z ` n ) = ( ( F oF - x ) ` n ) ) )
64	33, 62, 63	syl2anc 415	. . 3 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( z = ( F oF - x ) <-> A. n e. I ( z ` n ) = ( ( F oF - x ) ` n ) ) )
65	50, 57, 64	3bitr4d 220	. 2 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( x = ( F oF - z ) <-> z = ( F oF - x ) ) )
66	1, 4, 5, 65	f1o2d 6288	1 |- ( F e. D -> ( x e. S |-> ( F oF - x ) ) : S -1-1-onto-> S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   class class class wbr 4128   |-> cmpt 4190   `'ccnv 4771   "cima 4775   Fn wfn 5370   -->wf 5371   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6078   oFcof 6293   oRcofr 6294   ^m cmap 6915   Fincfn 7015   CCcc 8170   <_ cle 8354   - cmin 8490   NNcn 9286   NN0cn0 9545   ZZcz 9626

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-0id 8280  ax-rnegex 8281  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-ltadd 8288

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-of 6295  df-ofr 6296  df-1o 6680  df-er 6800  df-map 6917  df-en 7016  df-fin 7018  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-inn 9287  df-n0 9546  df-z 9627  df-uz 9904

This theorem is referenced by: (None)