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Theorem subfacp1lem4 24650
Description: Lemma for subfacp1 24653. 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 5655 . . . . . 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 24647 . . . 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 5629 . . 3  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  `' F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) )
14 f1ofn 5617 . . 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 5617 . . 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 24646 . . . . . . . 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 2456 . . . . . 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 3433 . . . . 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 24648 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  ( (  _I  |`  K ) `  x
) )
25 fvresi 5865 . . . . . . . . 9  |-  ( x  e.  K  ->  (
(  _I  |`  K ) `
 x )  =  x )
2625adantl 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
(  _I  |`  K ) `
 x )  =  x )
2724, 26eqtrd 2421 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  x )
2827fveq2d 5674 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  ( F `  x
) )
2928, 27eqtrd 2421 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  x )
30 vex 2904 . . . . . . 7  |-  x  e. 
_V
3130elpr 3777 . . . . . 6  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
3211simp2d 970 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  1
)  =  M )
3332fveq2d 5674 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  ( F `
 M ) )
3411simp3d 971 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  =  1 )
3533, 34eqtrd 2421 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  1 )
36 fveq2 5670 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
3736fveq2d 5674 . . . . . . . . . 10  |-  ( x  =  1  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  1 )
) )
38 id 20 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =  1 )
3937, 38eqeq12d 2403 . . . . . . . . 9  |-  ( x  =  1  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  1
) )  =  1 ) )
4035, 39syl5ibrcom 214 . . . . . . . 8  |-  ( ph  ->  ( x  =  1  ->  ( F `  ( F `  x ) )  =  x ) )
4134fveq2d 5674 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  M )
)  =  ( F `
 1 ) )
4241, 32eqtrd 2421 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  M )
)  =  M )
43 fveq2 5670 . . . . . . . . . . 11  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
4443fveq2d 5674 . . . . . . . . . 10  |-  ( x  =  M  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  M )
) )
45 id 20 . . . . . . . . . 10  |-  ( x  =  M  ->  x  =  M )
4644, 45eqeq12d 2403 . . . . . . . . 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 5616 . . . . . 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 5811 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  x )  e.  ( 1 ... ( N  +  1 ) ) )
57 f1ocnvfv 5957 . . . 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 5771 1  |-  ( ph  ->  `' F  =  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2375    =/= wne 2552   A.wral 2651   {crab 2655   _Vcvv 2901    \ cdif 3262    u. cun 3263    i^i cin 3264   (/)c0 3573   {csn 3759   {cpr 3760   <.cop 3762    e. cmpt 4209    _I cid 4436   `'ccnv 4819    |` cres 4822    Fn wfn 5391   -->wf 5392   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   Fincfn 7047   1c1 8926    + caddc 8928    - cmin 9225   NNcn 9934   2c2 9983   NN0cn0 10155   ...cfz 10977   #chash 11547
This theorem is referenced by:  subfacp1lem5  24651
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-hash 11548
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