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Theorem subfacp1lem4 23729
Description: Lemma for subfacp1 23732. 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 5527 . . . . . 6  |-  (  _I  |`  K ) : K -1-1-onto-> K
109a1i 10 . . . . 5  |-  ( ph  ->  (  _I  |`  K ) : K -1-1-onto-> K )
111, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2a 23726 . . . 4  |-  ( ph  ->  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  ( F ` 
1 )  =  M  /\  ( F `  M )  =  1 ) )
1211simp1d 967 . . 3  |-  ( ph  ->  F : ( 1 ... ( N  + 
1 ) ) -1-1-onto-> ( 1 ... ( N  + 
1 ) ) )
13 f1ocnv 5501 . . 3  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  `' F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) )
14 f1ofn 5489 . . 3  |-  ( `' F : ( 1 ... ( N  + 
1 ) ) -1-1-onto-> ( 1 ... ( N  + 
1 ) )  ->  `' F  Fn  (
1 ... ( N  + 
1 ) ) )
1512, 13, 143syl 18 . 2  |-  ( ph  ->  `' F  Fn  (
1 ... ( N  + 
1 ) ) )
16 f1ofn 5489 . . 3  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  F  Fn  ( 1 ... ( N  +  1 ) ) )
1712, 16syl 15 . 2  |-  ( ph  ->  F  Fn  ( 1 ... ( N  + 
1 ) ) )
181, 2, 3, 4, 5, 6, 7subfacp1lem1 23725 . . . . . . . 8  |-  ( ph  ->  ( ( K  i^i  { 1 ,  M }
)  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 ) )  /\  ( # `  K )  =  ( N  -  1 ) ) )
1918simp2d 968 . . . . . . 7  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
2019eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( x  e.  ( K  u.  { 1 ,  M } )  <-> 
x  e.  ( 1 ... ( N  + 
1 ) ) ) )
2120biimpar 471 . . . . 5  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  x  e.  ( K  u.  {
1 ,  M }
) )
22 elun 3329 . . . . 5  |-  ( x  e.  ( K  u.  { 1 ,  M }
)  <->  ( x  e.  K  \/  x  e. 
{ 1 ,  M } ) )
2321, 22sylib 188 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  (
x  e.  K  \/  x  e.  { 1 ,  M } ) )
241, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2b 23727 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  ( (  _I  |`  K ) `  x
) )
25 fvresi 5727 . . . . . . . . 9  |-  ( x  e.  K  ->  (
(  _I  |`  K ) `
 x )  =  x )
2625adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
(  _I  |`  K ) `
 x )  =  x )
2724, 26eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  x )
2827fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  ( F `  x
) )
2928, 27eqtrd 2328 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  x )
30 vex 2804 . . . . . . 7  |-  x  e. 
_V
3130elpr 3671 . . . . . 6  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
3211simp2d 968 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  1
)  =  M )
3332fveq2d 5545 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  ( F `
 M ) )
3411simp3d 969 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  =  1 )
3533, 34eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  1 )
36 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
3736fveq2d 5545 . . . . . . . . . 10  |-  ( x  =  1  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  1 )
) )
38 id 19 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =  1 )
3937, 38eqeq12d 2310 . . . . . . . . 9  |-  ( x  =  1  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  1
) )  =  1 ) )
4035, 39syl5ibrcom 213 . . . . . . . 8  |-  ( ph  ->  ( x  =  1  ->  ( F `  ( F `  x ) )  =  x ) )
4134fveq2d 5545 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  M )
)  =  ( F `
 1 ) )
4241, 32eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  M )
)  =  M )
43 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
4443fveq2d 5545 . . . . . . . . . 10  |-  ( x  =  M  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  M )
) )
45 id 19 . . . . . . . . . 10  |-  ( x  =  M  ->  x  =  M )
4644, 45eqeq12d 2310 . . . . . . . . 9  |-  ( x  =  M  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  M
) )  =  M ) )
4742, 46syl5ibrcom 213 . . . . . . . 8  |-  ( ph  ->  ( x  =  M  ->  ( F `  ( F `  x ) )  =  x ) )
4840, 47jaod 369 . . . . . . 7  |-  ( ph  ->  ( ( x  =  1  \/  x  =  M )  ->  ( F `  ( F `  x ) )  =  x ) )
4948imp 418 . . . . . 6  |-  ( (
ph  /\  ( x  =  1  \/  x  =  M ) )  -> 
( F `  ( F `  x )
)  =  x )
5031, 49sylan2b 461 . . . . 5  |-  ( (
ph  /\  x  e.  { 1 ,  M }
)  ->  ( F `  ( F `  x
) )  =  x )
5129, 50jaodan 760 . . . 4  |-  ( (
ph  /\  ( x  e.  K  \/  x  e.  { 1 ,  M } ) )  -> 
( F `  ( F `  x )
)  =  x )
5223, 51syldan 456 . . 3  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  ( F `  x ) )  =  x )
5312adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) )
54 f1of 5488 . . . . . 6  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  F :
( 1 ... ( N  +  1 ) ) --> ( 1 ... ( N  +  1 ) ) )
5512, 54syl 15 . . . . 5  |-  ( ph  ->  F : ( 1 ... ( N  + 
1 ) ) --> ( 1 ... ( N  +  1 ) ) )
56 ffvelrn 5679 . . . . 5  |-  ( ( F : ( 1 ... ( N  + 
1 ) ) --> ( 1 ... ( N  +  1 ) )  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  x )  e.  ( 1 ... ( N  +  1 ) ) )
5755, 56sylan 457 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  x )  e.  ( 1 ... ( N  +  1 ) ) )
58 f1ocnvfv 5810 . . . 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 ) ) )
5953, 57, 58syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  (
( F `  ( F `  x )
)  =  x  -> 
( `' F `  x )  =  ( F `  x ) ) )
6052, 59mpd 14 . 2  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( `' F `  x )  =  ( F `  x ) )
6115, 17, 60eqfnfvd 5641 1  |-  ( ph  ->  `' F  =  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Fincfn 6879   1c1 8754    + caddc 8756    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ...cfz 10798   #chash 11353
This theorem is referenced by:  subfacp1lem5  23730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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