Theorem psrbagconf1o 14774

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 2231	. 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 14773	. 2 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) e. S )
5	2, 3	psrbagconcl 14773	. 2 |- ( ( F e. D /\ z e. S ) -> ( F oF - z ) e. S )
6	2	psrbagf 14766	. . . . . . . . 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 5790	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) e. NN0 )
9	3	ssrab3 3314	. . . . . . . . . . . 12 |- S C_ D
10	9	sseli 3224	. . . . . . . . . . 11 |- ( z e. S -> z e. D )
11	10	adantl 277	. . . . . . . . . 10 |- ( ( F e. D /\ z e. S ) -> z e. D )
12	2	psrbagf 14766	. . . . . . . . . 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 478	. . . . . . . 8 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> z : I --> NN0 )
15	14	ffvelcdmda 5790	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) e. NN0 )
16		simprl 531	. . . . . . . . . 10 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x e. S )
17	9, 16	sselid 3226	. . . . . . . . 9 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x e. D )
18	2	psrbagf 14766	. . . . . . . . 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 5790	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) e. NN0 )
21		nn0cn 9471	. . . . . . . 8 |- ( ( F ` n ) e. NN0 -> ( F ` n ) e. CC )
22		nn0cn 9471	. . . . . . . 8 |- ( ( z ` n ) e. NN0 -> ( z ` n ) e. CC )
23		nn0cn 9471	. . . . . . . 8 |- ( ( x ` n ) e. NN0 -> ( x ` n ) e. CC )
24		subsub23 8443	. . . . . . . 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 1316	. . . . . . 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 1274	. . . . . 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 2233	. . . . . 6 |- ( ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) <-> ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) )
28		eqcom 2233	. . . . . 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 5490	. . . . . . . 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 5490	. . . . . . . 8 |- ( ( F e. D /\ z e. S ) -> z Fn I )
33	32	adantrl 478	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> z Fn I )
34	19	ffnd 5490	. . . . . . . 8 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x Fn I )
35	16, 34	fndmexd 5534	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> I e. _V )
36		inidm 3418	. . . . . . 7 |- ( I i^i I ) = I
37		eqidd 2232	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) = ( F ` n ) )
38		eqidd 2232	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) = ( z ` n ) )
39	8	nn0zd 9661	. . . . . . . 8 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) e. ZZ )
40	15	nn0zd 9661	. . . . . . . 8 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) e. ZZ )
41	39, 40	zsubcld 9668	. . . . . . 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 6254	. . . . . 6 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F oF - z ) ` n ) = ( ( F ` n ) - ( z ` n ) ) )
43	42	eqeq2d 2243	. . . . 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 2232	. . . . . . 7 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) = ( x ` n ) )
45	20	nn0zd 9661	. . . . . . . 8 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) e. ZZ )
46	39, 45	zsubcld 9668	. . . . . . 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 6254	. . . . . 6 |- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F oF - x ) ` n ) = ( ( F ` n ) - ( x ` n ) ) )
48	47	eqeq2d 2243	. . . . 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 2529	. . 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 478	. . . . . . 7 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) e. S )
52	9, 51	sselid 3226	. . . . . 6 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) e. D )
53	2	psrbagf 14766	. . . . . 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 5490	. . . 4 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) Fn I )
56		eqfnfv 5753	. . . 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 411	. . 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 3226	. . . . . . 7 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) e. D )
59	2	psrbagf 14766	. . . . . . 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 5490	. . . . 5 |- ( ( F e. D /\ x e. S ) -> ( F oF - x ) Fn I )
62	61	adantrr 479	. . . 4 |- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - x ) Fn I )
63		eqfnfv 5753	. . . 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 411	. . 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 6238	1 |- ( F e. D -> ( x e. S |-> ( F oF - x ) ) : S -1-1-onto-> S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   _Vcvv 2803   class class class wbr 4093   |-> cmpt 4155   `'ccnv 4730   "cima 4734   Fn wfn 5328   -->wf 5329   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   oFcof 6242   oRcofr 6243   ^m cmap 6860   Fincfn 6952   CCcc 8090   <_ cle 8274   - cmin 8409   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-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  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-dc 843  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-nul 3497  df-if 3608  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-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  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-of 6244  df-ofr 6245  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-fin 6955  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  df-uz 9817

This theorem is referenced by: (None)