Theorem subfacp1lem4 28443

Description: Lemma for subfacp1 28446. 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 28440	. . . 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 28439	. . . . . . . 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 2537	. . . . . 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 3650	. . . . 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 28441	. . . . . . . 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 2508	. . . . . . 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 2508	. . . . 5 |- ( ( ph /\ x e. K ) -> ( F ` ( F ` x ) ) = x )
30		vex 3121	. . . . . . 7 |- x e. _V
31	30	elpr 4051	. . . . . 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 2508	. . . . . . . . 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 2489	. . . . . . . . 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 2508	. . . . . . . . 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 2489	. . . . . . . . 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 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 6183	. . . 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 1379   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2817   {crab 2821   _Vcvv 3118   \ cdif 3478   u. cun 3479   i^i cin 3480   (/)c0 3790   {csn 4033   {cpr 4035   <.cop 4039   |-> cmpt 4511   _I cid 4796   `'ccnv 5004   |` cres 5007   Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Fincfn 7528   1c1 9505   + caddc 9507   - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ...cfz 11684   #chash 12385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386

This theorem is referenced by:  subfacp1lem5  28444