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Theorem subfacp1lem4 24859
Description: Lemma for subfacp1 24862. 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
StepHypRef 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 5705 . . . . . 6  |-  (  _I  |`  K ) : K -1-1-onto-> K
109a1i 11 . . . . 5  |-  ( ph  ->  (  _I  |`  K ) : K -1-1-onto-> K )
111, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2a 24856 . . . 4  |-  ( ph  ->  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  ( F ` 
1 )  =  M  /\  ( F `  M )  =  1 ) )
1211simp1d 969 . . 3  |-  ( ph  ->  F : ( 1 ... ( N  + 
1 ) ) -1-1-onto-> ( 1 ... ( N  + 
1 ) ) )
13 f1ocnv 5679 . . 3  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  `' F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) )
14 f1ofn 5667 . . 3  |-  ( `' F : ( 1 ... ( N  + 
1 ) ) -1-1-onto-> ( 1 ... ( N  + 
1 ) )  ->  `' F  Fn  (
1 ... ( N  + 
1 ) ) )
1512, 13, 143syl 19 . 2  |-  ( ph  ->  `' F  Fn  (
1 ... ( N  + 
1 ) ) )
16 f1ofn 5667 . . 3  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  F  Fn  ( 1 ... ( N  +  1 ) ) )
1712, 16syl 16 . 2  |-  ( ph  ->  F  Fn  ( 1 ... ( N  + 
1 ) ) )
181, 2, 3, 4, 5, 6, 7subfacp1lem1 24855 . . . . . . . 8  |-  ( ph  ->  ( ( K  i^i  { 1 ,  M }
)  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 ) )  /\  ( # `  K )  =  ( N  -  1 ) ) )
1918simp2d 970 . . . . . . 7  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
2019eleq2d 2502 . . . . . 6  |-  ( ph  ->  ( x  e.  ( K  u.  { 1 ,  M } )  <-> 
x  e.  ( 1 ... ( N  + 
1 ) ) ) )
2120biimpar 472 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  x  e.  ( K  u.  {
1 ,  M }
) )
22 elun 3480 . . . . 5  |-  ( x  e.  ( K  u.  { 1 ,  M }
)  <->  ( x  e.  K  \/  x  e. 
{ 1 ,  M } ) )
2321, 22sylib 189 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  (
x  e.  K  \/  x  e.  { 1 ,  M } ) )
241, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2b 24857 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  ( (  _I  |`  K ) `  x
) )
25 fvresi 5916 . . . . . . . . 9  |-  ( x  e.  K  ->  (
(  _I  |`  K ) `
 x )  =  x )
2625adantl 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
(  _I  |`  K ) `
 x )  =  x )
2724, 26eqtrd 2467 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  x )
2827fveq2d 5724 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  ( F `  x
) )
2928, 27eqtrd 2467 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  x )
30 vex 2951 . . . . . . 7  |-  x  e. 
_V
3130elpr 3824 . . . . . 6  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
3211simp2d 970 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  1
)  =  M )
3332fveq2d 5724 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  ( F `
 M ) )
3411simp3d 971 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  =  1 )
3533, 34eqtrd 2467 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  1 )
36 fveq2 5720 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
3736fveq2d 5724 . . . . . . . . . 10  |-  ( x  =  1  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  1 )
) )
38 id 20 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =  1 )
3937, 38eqeq12d 2449 . . . . . . . . 9  |-  ( x  =  1  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  1
) )  =  1 ) )
4035, 39syl5ibrcom 214 . . . . . . . 8  |-  ( ph  ->  ( x  =  1  ->  ( F `  ( F `  x ) )  =  x ) )
4134fveq2d 5724 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  M )
)  =  ( F `
 1 ) )
4241, 32eqtrd 2467 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  M )
)  =  M )
43 fveq2 5720 . . . . . . . . . . 11  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
4443fveq2d 5724 . . . . . . . . . 10  |-  ( x  =  M  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  M )
) )
45 id 20 . . . . . . . . . 10  |-  ( x  =  M  ->  x  =  M )
4644, 45eqeq12d 2449 . . . . . . . . 9  |-  ( x  =  M  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  M
) )  =  M ) )
4742, 46syl5ibrcom 214 . . . . . . . 8  |-  ( ph  ->  ( x  =  M  ->  ( F `  ( F `  x ) )  =  x ) )
4840, 47jaod 370 . . . . . . 7  |-  ( ph  ->  ( ( x  =  1  \/  x  =  M )  ->  ( F `  ( F `  x ) )  =  x ) )
4948imp 419 . . . . . 6  |-  ( (
ph  /\  ( x  =  1  \/  x  =  M ) )  -> 
( F `  ( F `  x )
)  =  x )
5031, 49sylan2b 462 . . . . 5  |-  ( (
ph  /\  x  e.  { 1 ,  M }
)  ->  ( F `  ( F `  x
) )  =  x )
5129, 50jaodan 761 . . . 4  |-  ( (
ph  /\  ( x  e.  K  \/  x  e.  { 1 ,  M } ) )  -> 
( F `  ( F `  x )
)  =  x )
5223, 51syldan 457 . . 3  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  ( F `  x ) )  =  x )
5312adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) )
54 f1of 5666 . . . . . 6  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  F :
( 1 ... ( N  +  1 ) ) --> ( 1 ... ( N  +  1 ) ) )
5512, 54syl 16 . . . . 5  |-  ( ph  ->  F : ( 1 ... ( N  + 
1 ) ) --> ( 1 ... ( N  +  1 ) ) )
5655ffvelrnda 5862 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  x )  e.  ( 1 ... ( N  +  1 ) ) )
57 f1ocnvfv 6008 . . . 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 ) ) )
5853, 56, 57syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  (
( F `  ( F `  x )
)  =  x  -> 
( `' F `  x )  =  ( F `  x ) ) )
5952, 58mpd 15 . 2  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( `' F `  x )  =  ( F `  x ) )
6015, 17, 59eqfnfvd 5822 1  |-  ( ph  ->  `' F  =  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   {cpr 3807   <.cop 3809    e. cmpt 4258    _I cid 4485   `'ccnv 4869    |` cres 4872    Fn wfn 5441   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Fincfn 7101   1c1 8981    + caddc 8983    - cmin 9281   NNcn 9990   2c2 10039   NN0cn0 10211   ...cfz 11033   #chash 11608
This theorem is referenced by:  subfacp1lem5  24860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-hash 11609
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