Theorem subfacp1lem4 31165

Description: Lemma for subfacp1 31168. The function F, which swaps 1 with M and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypotheses

Ref	Expression
derang.d	|- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) )
subfac.n	|- S = ( n e. NN0 |-> ( D ` ( 1 ... n ) ) )
subfacp1lem.a	|- A = { f | ( f : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ A. y e. ( 1 ... ( N + 1 ) ) ( f ` y ) =/= y ) }
subfacp1lem1.n	|- ( ph -> N e. NN )
subfacp1lem1.m	|- ( ph -> M e. ( 2 ... ( N + 1 ) ) )
subfacp1lem1.x	|- M e. _V
subfacp1lem1.k	|- K = ( ( 2 ... ( N + 1 ) ) \ { M } )
subfacp1lem5.b	|- B = { g e. A | ( ( g ` 1 ) = M /\ ( g ` M ) =/= 1 ) }
subfacp1lem5.f	|- F = ( ( _I |` K ) u. { <. 1 , M >. , <. M , 1 >. } )

Assertion

Ref	Expression
subfacp1lem4	|- ( ph -> `' F = F )

Distinct variable groups:   f, g, n, x, y, A   f, F, g, x, y   f, N, g, n, x, y   B, f, g, x, y   ph, x, y   D, n   f, K, n, x, y   f, M, g, x, y   S, n, x, y

Allowed substitution hints:   ph( f, g, n)   B( n)   D( x, y, f, g)   S( f, g)   F( n)   K( g)   M( n)

Proof of Theorem subfacp1lem4

Step	Hyp	Ref	Expression
1		derang.d	. . . . 5 |- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) )
2		subfac.n	. . . . 5 |- S = ( n e. NN0 |-> ( D ` ( 1 ... n ) ) )
3		subfacp1lem.a	. . . . 5 |- A = { f | ( f : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ A. y e. ( 1 ... ( N + 1 ) ) ( f ` y ) =/= y ) }
4		subfacp1lem1.n	. . . . 5 |- ( ph -> N e. NN )
5		subfacp1lem1.m	. . . . 5 |- ( ph -> M e. ( 2 ... ( N + 1 ) ) )
6		subfacp1lem1.x	. . . . 5 |- M e. _V
7		subfacp1lem1.k	. . . . 5 |- K = ( ( 2 ... ( N + 1 ) ) \ { M } )
8		subfacp1lem5.f	. . . . 5 |- F = ( ( _I |` K ) u. { <. 1 , M >. , <. M , 1 >. } )
9		f1oi 6174	. . . . . 6 |- ( _I |` K ) : K -1-1-onto-> K
10	9	a1i 11	. . . . 5 |- ( ph -> ( _I |` K ) : K -1-1-onto-> K )
11	1, 2, 3, 4, 5, 6, 7, 8, 10	subfacp1lem2a 31162	. . . 4 |- ( ph -> ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ ( F ` 1 ) = M /\ ( F ` M ) = 1 ) )
12	11	simp1d 1073	. . 3 |- ( ph -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
13		f1ocnv 6149	. . 3 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
14		f1ofn 6138	. . 3 |- ( `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F Fn ( 1 ... ( N + 1 ) ) )
15	12, 13, 14	3syl 18	. 2 |- ( ph -> `' F Fn ( 1 ... ( N + 1 ) ) )
16		f1ofn 6138	. . 3 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> F Fn ( 1 ... ( N + 1 ) ) )
17	12, 16	syl 17	. 2 |- ( ph -> F Fn ( 1 ... ( N + 1 ) ) )
18	1, 2, 3, 4, 5, 6, 7	subfacp1lem1 31161	. . . . . . . 8 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )
19	18	simp2d 1074	. . . . . . 7 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
20	19	eleq2d 2687	. . . . . 6 |- ( ph -> ( x e. ( K u. { 1 , M } ) <-> x e. ( 1 ... ( N + 1 ) ) ) )
21	20	biimpar 502	. . . . 5 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> x e. ( K u. { 1 , M } ) )
22		elun 3753	. . . . 5 |- ( x e. ( K u. { 1 , M } ) <-> ( x e. K \/ x e. { 1 , M } ) )
23	21, 22	sylib 208	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( x e. K \/ x e. { 1 , M } ) )
24	1, 2, 3, 4, 5, 6, 7, 8, 10	subfacp1lem2b 31163	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( F ` x ) = ( ( _I |` K ) ` x ) )
25		fvresi 6439	. . . . . . . . 9 |- ( x e. K -> ( ( _I |` K ) ` x ) = x )
26	25	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( _I |` K ) ` x ) = x )
27	24, 26	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( F ` x ) = x )
28	27	fveq2d 6195	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = ( F ` x ) )
29	28, 27	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = x )
30		vex 3203	. . . . . . 7 |- x e. _V
31	30	elpr 4198	. . . . . 6 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
32	11	simp2d 1074	. . . . . . . . . . 11 |- ( ph -> ( F ` 1 ) = M )
33	32	fveq2d 6195	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` 1 ) ) = ( F ` M ) )
34	11	simp3d 1075	. . . . . . . . . 10 |- ( ph -> ( F ` M ) = 1 )
35	33, 34	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` 1 ) ) = 1 )
36		fveq2 6191	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
37	36	fveq2d 6195	. . . . . . . . . 10 |- ( x = 1 -> ( F ` ( F ` x ) ) = ( F ` ( F ` 1 ) ) )
38		id 22	. . . . . . . . . 10 |- ( x = 1 -> x = 1 )
39	37, 38	eqeq12d 2637	. . . . . . . . 9 |- ( x = 1 -> ( ( F ` ( F ` x ) ) = x <-> ( F ` ( F ` 1 ) ) = 1 ) )
40	35, 39	syl5ibrcom 237	. . . . . . . 8 |- ( ph -> ( x = 1 -> ( F ` ( F ` x ) ) = x ) )
41	34	fveq2d 6195	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` M ) ) = ( F ` 1 ) )
42	41, 32	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` M ) ) = M )
43		fveq2 6191	. . . . . . . . . . 11 |- ( x = M -> ( F ` x ) = ( F ` M ) )
44	43	fveq2d 6195	. . . . . . . . . 10 |- ( x = M -> ( F ` ( F ` x ) ) = ( F ` ( F ` M ) ) )
45		id 22	. . . . . . . . . 10 |- ( x = M -> x = M )
46	44, 45	eqeq12d 2637	. . . . . . . . 9 |- ( x = M -> ( ( F ` ( F ` x ) ) = x <-> ( F ` ( F ` M ) ) = M ) )
47	42, 46	syl5ibrcom 237	. . . . . . . 8 |- ( ph -> ( x = M -> ( F ` ( F ` x ) ) = x ) )
48	40, 47	jaod 395	. . . . . . 7 |- ( ph -> ( ( x = 1 \/ x = M ) -> ( F ` ( F ` x ) ) = x ) )
49	48	imp 445	. . . . . 6 |- ( ( ph /\ ( x = 1 \/ x = M ) ) -> ( F ` ( F ` x ) ) = x )
50	31, 49	sylan2b 492	. . . . 5 |- ( ( ph /\ x e. { 1 , M } ) -> ( F ` ( F ` x ) ) = x )
51	29, 50	jaodan 826	. . . 4 |- ( ( ph /\ ( x e. K \/ x e. { 1 , M } ) ) -> ( F ` ( F ` x ) ) = x )
52	23, 51	syldan 487	. . 3 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` ( F ` x ) ) = x )
53	12	adantr 481	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
54		f1of 6137	. . . . . 6 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> F : ( 1 ... ( N + 1 ) ) --> ( 1 ... ( N + 1 ) ) )
55	12, 54	syl 17	. . . . 5 |- ( ph -> F : ( 1 ... ( N + 1 ) ) --> ( 1 ... ( N + 1 ) ) )
56	55	ffvelrnda 6359	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` x ) e. ( 1 ... ( N + 1 ) ) )
57		f1ocnvfv 6534	. . . 4 |- ( ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ ( F ` x ) e. ( 1 ... ( N + 1 ) ) ) -> ( ( F ` ( F ` x ) ) = x -> ( `' F ` x ) = ( F ` x ) ) )
58	53, 56, 57	syl2anc 693	. . 3 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( ( F ` ( F ` x ) ) = x -> ( `' F ` x ) = ( F ` x ) ) )
59	52, 58	mpd 15	. 2 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( `' F ` x ) = ( F ` x ) )
60	15, 17, 59	eqfnfvd 6314	1 |- ( ph -> `' F = F )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  subfacp1lem5  31166