Theorem subfacp1lem4 28824

Description: Lemma for subfacp1 28827. 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 5857	. . . . . 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 28821	. . . 4 |- ( ph -> ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ ( F ` 1 ) = M /\ ( F ` M ) = 1 ) )
12	11	simp1d 1008	. . 3 |- ( ph -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
13		f1ocnv 5834	. . 3 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
14		f1ofn 5823	. . 3 |- ( `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F Fn ( 1 ... ( N + 1 ) ) )
15	12, 13, 14	3syl 20	. 2 |- ( ph -> `' F Fn ( 1 ... ( N + 1 ) ) )
16		f1ofn 5823	. . 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 28820	. . . . . . . 8 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )
19	18	simp2d 1009	. . . . . . 7 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
20	19	eleq2d 2527	. . . . . 6 |- ( ph -> ( x e. ( K u. { 1 , M } ) <-> x e. ( 1 ... ( N + 1 ) ) ) )
21	20	biimpar 485	. . . . 5 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> x e. ( K u. { 1 , M } ) )
22		elun 3641	. . . . 5 |- ( x e. ( K u. { 1 , M } ) <-> ( x e. K \/ x e. { 1 , M } ) )
23	21, 22	sylib 196	. . . 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 28822	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( F ` x ) = ( ( _I |` K ) ` x ) )
25		fvresi 6098	. . . . . . . . 9 |- ( x e. K -> ( ( _I |` K ) ` x ) = x )
26	25	adantl 466	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( _I |` K ) ` x ) = x )
27	24, 26	eqtrd 2498	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( F ` x ) = x )
28	27	fveq2d 5876	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = ( F ` x ) )
29	28, 27	eqtrd 2498	. . . . 5 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = x )
30		vex 3112	. . . . . . 7 |- x e. _V
31	30	elpr 4050	. . . . . 6 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
32	11	simp2d 1009	. . . . . . . . . . 11 |- ( ph -> ( F ` 1 ) = M )
33	32	fveq2d 5876	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` 1 ) ) = ( F ` M ) )
34	11	simp3d 1010	. . . . . . . . . 10 |- ( ph -> ( F ` M ) = 1 )
35	33, 34	eqtrd 2498	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` 1 ) ) = 1 )
36		fveq2 5872	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
37	36	fveq2d 5876	. . . . . . . . . 10 |- ( x = 1 -> ( F ` ( F ` x ) ) = ( F ` ( F ` 1 ) ) )
38		id 22	. . . . . . . . . 10 |- ( x = 1 -> x = 1 )
39	37, 38	eqeq12d 2479	. . . . . . . . 9 |- ( x = 1 -> ( ( F ` ( F ` x ) ) = x <-> ( F ` ( F ` 1 ) ) = 1 ) )
40	35, 39	syl5ibrcom 222	. . . . . . . 8 |- ( ph -> ( x = 1 -> ( F ` ( F ` x ) ) = x ) )
41	34	fveq2d 5876	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` M ) ) = ( F ` 1 ) )
42	41, 32	eqtrd 2498	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` M ) ) = M )
43		fveq2 5872	. . . . . . . . . . 11 |- ( x = M -> ( F ` x ) = ( F ` M ) )
44	43	fveq2d 5876	. . . . . . . . . 10 |- ( x = M -> ( F ` ( F ` x ) ) = ( F ` ( F ` M ) ) )
45		id 22	. . . . . . . . . 10 |- ( x = M -> x = M )
46	44, 45	eqeq12d 2479	. . . . . . . . 9 |- ( x = M -> ( ( F ` ( F ` x ) ) = x <-> ( F ` ( F ` M ) ) = M ) )
47	42, 46	syl5ibrcom 222	. . . . . . . 8 |- ( ph -> ( x = M -> ( F ` ( F ` x ) ) = x ) )
48	40, 47	jaod 380	. . . . . . 7 |- ( ph -> ( ( x = 1 \/ x = M ) -> ( F ` ( F ` x ) ) = x ) )
49	48	imp 429	. . . . . 6 |- ( ( ph /\ ( x = 1 \/ x = M ) ) -> ( F ` ( F ` x ) ) = x )
50	31, 49	sylan2b 475	. . . . 5 |- ( ( ph /\ x e. { 1 , M } ) -> ( F ` ( F ` x ) ) = x )
51	29, 50	jaodan 785	. . . 4 |- ( ( ph /\ ( x e. K \/ x e. { 1 , M } ) ) -> ( F ` ( F ` x ) ) = x )
52	23, 51	syldan 470	. . 3 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` ( F ` x ) ) = x )
53	12	adantr 465	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
54		f1of 5822	. . . . . 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 6032	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` x ) e. ( 1 ... ( N + 1 ) ) )
57		f1ocnvfv 6185	. . . 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 661	. . 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 5985	1 |- ( ph -> `' F = F )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   {cab 2442   =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   \ cdif 3468   u. cun 3469   i^i cin 3470   (/)c0 3793   {csn 4032   {cpr 4034   <.cop 4038   |-> cmpt 4515   _I cid 4799   `'ccnv 5007   |` cres 5010   Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   Fincfn 7535   1c1 9510   + caddc 9512   - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ...cfz 11697   #chash 12408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409

This theorem is referenced by:  subfacp1lem5  28825