Theorem subfacp1lem4 24822

Description: Lemma for subfacp1 24825. 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 5672	. . . . . 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 24819	. . . 4 |- ( ph -> ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ ( F ` 1 ) = M /\ ( F ` M ) = 1 ) )
12	11	simp1d 969	. . 3 |- ( ph -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
13		f1ocnv 5646	. . 3 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
14		f1ofn 5634	. . 3 |- ( `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F Fn ( 1 ... ( N + 1 ) ) )
15	12, 13, 14	3syl 19	. 2 |- ( ph -> `' F Fn ( 1 ... ( N + 1 ) ) )
16		f1ofn 5634	. . 3 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> F Fn ( 1 ... ( N + 1 ) ) )
17	12, 16	syl 16	. 2 |- ( ph -> F Fn ( 1 ... ( N + 1 ) ) )
18	1, 2, 3, 4, 5, 6, 7	subfacp1lem1 24818	. . . . . . . 8 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )
19	18	simp2d 970	. . . . . . 7 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
20	19	eleq2d 2471	. . . . . 6 |- ( ph -> ( x e. ( K u. { 1 , M } ) <-> x e. ( 1 ... ( N + 1 ) ) ) )
21	20	biimpar 472	. . . . 5 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> x e. ( K u. { 1 , M } ) )
22		elun 3448	. . . . 5 |- ( x e. ( K u. { 1 , M } ) <-> ( x e. K \/ x e. { 1 , M } ) )
23	21, 22	sylib 189	. . . 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 24820	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( F ` x ) = ( ( _I |` K ) ` x ) )
25		fvresi 5883	. . . . . . . . 9 |- ( x e. K -> ( ( _I |` K ) ` x ) = x )
26	25	adantl 453	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( _I |` K ) ` x ) = x )
27	24, 26	eqtrd 2436	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( F ` x ) = x )
28	27	fveq2d 5691	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = ( F ` x ) )
29	28, 27	eqtrd 2436	. . . . 5 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = x )
30		vex 2919	. . . . . . 7 |- x e. _V
31	30	elpr 3792	. . . . . 6 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
32	11	simp2d 970	. . . . . . . . . . 11 |- ( ph -> ( F ` 1 ) = M )
33	32	fveq2d 5691	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` 1 ) ) = ( F ` M ) )
34	11	simp3d 971	. . . . . . . . . 10 |- ( ph -> ( F ` M ) = 1 )
35	33, 34	eqtrd 2436	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` 1 ) ) = 1 )
36		fveq2 5687	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
37	36	fveq2d 5691	. . . . . . . . . 10 |- ( x = 1 -> ( F ` ( F ` x ) ) = ( F ` ( F ` 1 ) ) )
38		id 20	. . . . . . . . . 10 |- ( x = 1 -> x = 1 )
39	37, 38	eqeq12d 2418	. . . . . . . . 9 |- ( x = 1 -> ( ( F ` ( F ` x ) ) = x <-> ( F ` ( F ` 1 ) ) = 1 ) )
40	35, 39	syl5ibrcom 214	. . . . . . . 8 |- ( ph -> ( x = 1 -> ( F ` ( F ` x ) ) = x ) )
41	34	fveq2d 5691	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` M ) ) = ( F ` 1 ) )
42	41, 32	eqtrd 2436	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` M ) ) = M )
43		fveq2 5687	. . . . . . . . . . 11 |- ( x = M -> ( F ` x ) = ( F ` M ) )
44	43	fveq2d 5691	. . . . . . . . . 10 |- ( x = M -> ( F ` ( F ` x ) ) = ( F ` ( F ` M ) ) )
45		id 20	. . . . . . . . . 10 |- ( x = M -> x = M )
46	44, 45	eqeq12d 2418	. . . . . . . . 9 |- ( x = M -> ( ( F ` ( F ` x ) ) = x <-> ( F ` ( F ` M ) ) = M ) )
47	42, 46	syl5ibrcom 214	. . . . . . . 8 |- ( ph -> ( x = M -> ( F ` ( F ` x ) ) = x ) )
48	40, 47	jaod 370	. . . . . . 7 |- ( ph -> ( ( x = 1 \/ x = M ) -> ( F ` ( F ` x ) ) = x ) )
49	48	imp 419	. . . . . 6 |- ( ( ph /\ ( x = 1 \/ x = M ) ) -> ( F ` ( F ` x ) ) = x )
50	31, 49	sylan2b 462	. . . . 5 |- ( ( ph /\ x e. { 1 , M } ) -> ( F ` ( F ` x ) ) = x )
51	29, 50	jaodan 761	. . . 4 |- ( ( ph /\ ( x e. K \/ x e. { 1 , M } ) ) -> ( F ` ( F ` x ) ) = x )
52	23, 51	syldan 457	. . 3 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` ( F ` x ) ) = x )
53	12	adantr 452	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
54		f1of 5633	. . . . . 6 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> F : ( 1 ... ( N + 1 ) ) --> ( 1 ... ( N + 1 ) ) )
55	12, 54	syl 16	. . . . 5 |- ( ph -> F : ( 1 ... ( N + 1 ) ) --> ( 1 ... ( N + 1 ) ) )
56	55	ffvelrnda 5829	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` x ) e. ( 1 ... ( N + 1 ) ) )
57		f1ocnvfv 5975	. . . 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 643	. . 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 5789	1 |- ( ph -> `' F = F )

Syntax hints:   -> wi 4   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   {cab 2390   =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916   \ cdif 3277   u. cun 3278   i^i cin 3279   (/)c0 3588   {csn 3774   {cpr 3775   <.cop 3777   e. cmpt 4226   _I cid 4453   `'ccnv 4836   |` cres 4839   Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   1c1 8947   + caddc 8949   - cmin 9247   NNcn 9956   2c2 10005   NN0cn0 10177   ...cfz 10999   #chash 11573

This theorem is referenced by:  subfacp1lem5  24823

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574