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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
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 5857 . . . . . 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 28440 . . . 4  |-  ( ph  ->  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  ( F ` 
1 )  =  M  /\  ( F `  M )  =  1 ) )
1211simp1d 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 ) ) )
1512, 13, 143syl 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 ) ) )
1712, 16syl 16 . 2  |-  ( ph  ->  F  Fn  ( 1 ... ( N  + 
1 ) ) )
181, 2, 3, 4, 5, 6, 7subfacp1lem1 28439 . . . . . . . 8  |-  ( ph  ->  ( ( K  i^i  { 1 ,  M }
)  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 ) )  /\  ( # `  K )  =  ( N  -  1 ) ) )
1918simp2d 1009 . . . . . . 7  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
2019eleq2d 2537 . . . . . 6  |-  ( ph  ->  ( x  e.  ( K  u.  { 1 ,  M } )  <-> 
x  e.  ( 1 ... ( N  + 
1 ) ) ) )
2120biimpar 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 } ) )
2321, 22sylib 196 . . . 4  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  (
x  e.  K  \/  x  e.  { 1 ,  M } ) )
241, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2b 28441 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  ( (  _I  |`  K ) `  x
) )
25 fvresi 6098 . . . . . . . . 9  |-  ( x  e.  K  ->  (
(  _I  |`  K ) `
 x )  =  x )
2625adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
(  _I  |`  K ) `
 x )  =  x )
2724, 26eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  x )  =  x )
2827fveq2d 5876 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  ( F `  x
) )
2928, 27eqtrd 2508 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  ( F `  ( F `  x ) )  =  x )
30 vex 3121 . . . . . . 7  |-  x  e. 
_V
3130elpr 4051 . . . . . 6  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
3211simp2d 1009 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  1
)  =  M )
3332fveq2d 5876 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  ( F `
 M ) )
3411simp3d 1010 . . . . . . . . . 10  |-  ( ph  ->  ( F `  M
)  =  1 )
3533, 34eqtrd 2508 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  1 )
)  =  1 )
36 fveq2 5872 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
3736fveq2d 5876 . . . . . . . . . 10  |-  ( x  =  1  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  1 )
) )
38 id 22 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =  1 )
3937, 38eqeq12d 2489 . . . . . . . . 9  |-  ( x  =  1  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  1
) )  =  1 ) )
4035, 39syl5ibrcom 222 . . . . . . . 8  |-  ( ph  ->  ( x  =  1  ->  ( F `  ( F `  x ) )  =  x ) )
4134fveq2d 5876 . . . . . . . . . 10  |-  ( ph  ->  ( F `  ( F `  M )
)  =  ( F `
 1 ) )
4241, 32eqtrd 2508 . . . . . . . . 9  |-  ( ph  ->  ( F `  ( F `  M )
)  =  M )
43 fveq2 5872 . . . . . . . . . . 11  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
4443fveq2d 5876 . . . . . . . . . 10  |-  ( x  =  M  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  M )
) )
45 id 22 . . . . . . . . . 10  |-  ( x  =  M  ->  x  =  M )
4644, 45eqeq12d 2489 . . . . . . . . 9  |-  ( x  =  M  ->  (
( F `  ( F `  x )
)  =  x  <->  ( F `  ( F `  M
) )  =  M ) )
4742, 46syl5ibrcom 222 . . . . . . . 8  |-  ( ph  ->  ( x  =  M  ->  ( F `  ( F `  x ) )  =  x ) )
4840, 47jaod 380 . . . . . . 7  |-  ( ph  ->  ( ( x  =  1  \/  x  =  M )  ->  ( F `  ( F `  x ) )  =  x ) )
4948imp 429 . . . . . 6  |-  ( (
ph  /\  ( x  =  1  \/  x  =  M ) )  -> 
( F `  ( F `  x )
)  =  x )
5031, 49sylan2b 475 . . . . 5  |-  ( (
ph  /\  x  e.  { 1 ,  M }
)  ->  ( F `  ( F `  x
) )  =  x )
5129, 50jaodan 783 . . . 4  |-  ( (
ph  /\  ( x  e.  K  \/  x  e.  { 1 ,  M } ) )  -> 
( F `  ( F `  x )
)  =  x )
5223, 51syldan 470 . . 3  |-  ( (
ph  /\  x  e.  ( 1 ... ( N  +  1 ) ) )  ->  ( F `  ( F `  x ) )  =  x )
5312adantr 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 ) ) )
5512, 54syl 16 . . . . 5  |-  ( ph  ->  F : ( 1 ... ( N  + 
1 ) ) --> ( 1 ... ( N  +  1 ) ) )
5655ffvelrnda 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 ) ) )
5853, 56, 57syl2anc 661 . . 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 5985 1  |-  ( ph  ->  `' F  =  F
)
Colors of variables: wff setvar class
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
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