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Theorem subfacp1lem4 23714
Description: Lemma for subfacp1 23717. 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 5511 . . . . . 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 23711 . . . 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 5485 . . 3  |-  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  ->  `' F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) )
14 f1ofn 5473 . . 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 5473 . . 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 23710 . . . . . . . 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 2350 . . . . . 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 3316 . . . . 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 23712 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  ( (  _I  |`  K ) `  x
) )
25 fvresi 5711 . . . . . . . . 9  |-  ( x  e.  K  ->  (
(  _I  |`  K ) `
 x )  =  x )
2625adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
(  _I  |`  K ) `
 x )  =  x )
2724, 26eqtrd 2315 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  x )
2827fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  ( F `  x
) )
2928, 27eqtrd 2315 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  x )
30 vex 2791 . . . . . . 7  |-  x  e. 
_V
3130elpr 3658 . . . . . 6  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
3211simp2d 968 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  1
)  =  M )
3332fveq2d 5529 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  ( F `
 M ) )
3411simp3d 969 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  =  1 )
3533, 34eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  1 )
36 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
3736fveq2d 5529 . . . . . . . . . 10  |-  ( x  =  1  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  1 )
) )
38 id 19 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =  1 )
3937, 38eqeq12d 2297 . . . . . . . . 9  |-  ( x  =  1  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  1
) )  =  1 ) )
4035, 39syl5ibrcom 213 . . . . . . . 8  |-  ( ph  ->  ( x  =  1  ->  ( F `  ( F `  x ) )  =  x ) )
4134fveq2d 5529 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  M )
)  =  ( F `
 1 ) )
4241, 32eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  M )
)  =  M )
43 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
4443fveq2d 5529 . . . . . . . . . 10  |-  ( x  =  M  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  M )
) )
45 id 19 . . . . . . . . . 10  |-  ( x  =  M  ->  x  =  M )
4644, 45eqeq12d 2297 . . . . . . . . 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 5472 . . . . . 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 5663 . . . . 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 5794 . . . 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 5625 1  |-  ( ph  ->  `' F  =  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   {cpr 3641   <.cop 3643    e. cmpt 4077    _I cid 4304   `'ccnv 4688    |` cres 4691    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Fincfn 6863   1c1 8738    + caddc 8740    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ...cfz 10782   #chash 11337
This theorem is referenced by:  subfacp1lem5  23715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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