Theorem subfacp1lem4 27076

Description: Lemma for subfacp1 27079. 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 5681	. . . . . 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 27073	. . . 4 |- ( ph -> ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) /\ ( F ` 1 ) = M /\ ( F ` M ) = 1 ) )
12	11	simp1d 1000	. . 3 |- ( ph -> F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
13		f1ocnv 5658	. . 3 |- ( F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) -> `' F : ( 1 ... ( N + 1 ) ) -1-1-onto-> ( 1 ... ( N + 1 ) ) )
14		f1ofn 5647	. . 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 5647	. . 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 27072	. . . . . . . 8 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )
19	18	simp2d 1001	. . . . . . 7 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
20	19	eleq2d 2510	. . . . . 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 3502	. . . . 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 27074	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( F ` x ) = ( ( _I |` K ) ` x ) )
25		fvresi 5909	. . . . . . . . 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 2475	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( F ` x ) = x )
28	27	fveq2d 5700	. . . . . 6 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = ( F ` x ) )
29	28, 27	eqtrd 2475	. . . . 5 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = x )
30		vex 2980	. . . . . . 7 |- x e. _V
31	30	elpr 3900	. . . . . 6 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
32	11	simp2d 1001	. . . . . . . . . . 11 |- ( ph -> ( F ` 1 ) = M )
33	32	fveq2d 5700	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` 1 ) ) = ( F ` M ) )
34	11	simp3d 1002	. . . . . . . . . 10 |- ( ph -> ( F ` M ) = 1 )
35	33, 34	eqtrd 2475	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` 1 ) ) = 1 )
36		fveq2 5696	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
37	36	fveq2d 5700	. . . . . . . . . 10 |- ( x = 1 -> ( F ` ( F ` x ) ) = ( F ` ( F ` 1 ) ) )
38		id 22	. . . . . . . . . 10 |- ( x = 1 -> x = 1 )
39	37, 38	eqeq12d 2457	. . . . . . . . 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 5700	. . . . . . . . . 10 |- ( ph -> ( F ` ( F ` M ) ) = ( F ` 1 ) )
42	41, 32	eqtrd 2475	. . . . . . . . 9 |- ( ph -> ( F ` ( F ` M ) ) = M )
43		fveq2 5696	. . . . . . . . . . 11 |- ( x = M -> ( F ` x ) = ( F ` M ) )
44	43	fveq2d 5700	. . . . . . . . . 10 |- ( x = M -> ( F ` ( F ` x ) ) = ( F ` ( F ` M ) ) )
45		id 22	. . . . . . . . . 10 |- ( x = M -> x = M )
46	44, 45	eqeq12d 2457	. . . . . . . . 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 783	. . . 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 5646	. . . . . 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 5848	. . . 4 |- ( ( ph /\ x e. ( 1 ... ( N + 1 ) ) ) -> ( F ` x ) e. ( 1 ... ( N + 1 ) ) )
57		f1ocnvfv 5990	. . . 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 5805	1 |- ( ph -> `' F = F )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   {cab 2429   =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977   \ cdif 3330   u. cun 3331   i^i cin 3332   (/)c0 3642   {csn 3882   {cpr 3884   <.cop 3888   e. cmpt 4355   _I cid 4636   `'ccnv 4844   |` cres 4847   Fn wfn 5418   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   Fincfn 7315   1c1 9288   + caddc 9290   - cmin 9600   NNcn 10327   2c2 10376   NN0cn0 10584   ...cfz 11442   #chash 12108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-hash 12109

This theorem is referenced by:  subfacp1lem5  27077