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Theorem pmtrfconj 26730
Description: Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrfconj  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  (
( G  o.  F
)  o.  `' G
)  e.  R )

Proof of Theorem pmtrfconj
StepHypRef Expression
1 pmtrrn.t . . . . 5  |-  T  =  (pmTrsp `  D )
2 pmtrrn.r . . . . 5  |-  R  =  ran  T
31, 2pmtrfb 26729 . . . 4  |-  ( F  e.  R  <->  ( D  e.  _V  /\  F : D
-1-1-onto-> D  /\  dom  ( F 
\  _I  )  ~~  2o ) )
43simp1bi 970 . . 3  |-  ( F  e.  R  ->  D  e.  _V )
54adantr 451 . 2  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  D  e.  _V )
6 simpr 447 . . . 4  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  G : D -1-1-onto-> D )
71, 2pmtrff1o 26727 . . . . 5  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
87adantr 451 . . . 4  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  F : D -1-1-onto-> D )
9 f1oco 5576 . . . 4  |-  ( ( G : D -1-1-onto-> D  /\  F : D -1-1-onto-> D )  ->  ( G  o.  F ) : D -1-1-onto-> D )
106, 8, 9syl2anc 642 . . 3  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  ( G  o.  F ) : D -1-1-onto-> D )
11 f1ocnv 5565 . . . 4  |-  ( G : D -1-1-onto-> D  ->  `' G : D -1-1-onto-> D )
1211adantl 452 . . 3  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  `' G : D -1-1-onto-> D )
13 f1oco 5576 . . 3  |-  ( ( ( G  o.  F
) : D -1-1-onto-> D  /\  `' G : D -1-1-onto-> D )  ->  ( ( G  o.  F )  o.  `' G ) : D -1-1-onto-> D
)
1410, 12, 13syl2anc 642 . 2  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  (
( G  o.  F
)  o.  `' G
) : D -1-1-onto-> D )
15 f1of 5552 . . . . . . 7  |-  ( F : D -1-1-onto-> D  ->  F : D
--> D )
167, 15syl 15 . . . . . 6  |-  ( F  e.  R  ->  F : D --> D )
1716adantr 451 . . . . 5  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  F : D --> D )
18 f1omvdconj 26712 . . . . 5  |-  ( ( F : D --> D  /\  G : D -1-1-onto-> D )  ->  dom  ( ( ( G  o.  F )  o.  `' G )  \  _I  )  =  ( G " dom  ( F  \  _I  ) ) )
1917, 6, 18syl2anc 642 . . . 4  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  dom  ( ( ( G  o.  F )  o.  `' G )  \  _I  )  =  ( G " dom  ( F  \  _I  ) ) )
20 f1of1 5551 . . . . . 6  |-  ( G : D -1-1-onto-> D  ->  G : D -1-1-> D )
2120adantl 452 . . . . 5  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  G : D -1-1-> D )
22 difss 3379 . . . . . . . 8  |-  ( F 
\  _I  )  C_  F
23 dmss 4957 . . . . . . . 8  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
2422, 23ax-mp 8 . . . . . . 7  |-  dom  ( F  \  _I  )  C_  dom  F
25 fdm 5473 . . . . . . 7  |-  ( F : D --> D  ->  dom  F  =  D )
2624, 25syl5sseq 3302 . . . . . 6  |-  ( F : D --> D  ->  dom  ( F  \  _I  )  C_  D )
2717, 26syl 15 . . . . 5  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  dom  ( F  \  _I  )  C_  D )
28 ssexg 4239 . . . . . 6  |-  ( ( dom  ( F  \  _I  )  C_  D  /\  D  e.  _V )  ->  dom  ( F  \  _I  )  e.  _V )
2927, 5, 28syl2anc 642 . . . . 5  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  dom  ( F  \  _I  )  e.  _V )
30 f1imaeng 7006 . . . . 5  |-  ( ( G : D -1-1-> D  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  e.  _V )  ->  ( G " dom  ( F  \  _I  )
)  ~~  dom  ( F 
\  _I  ) )
3121, 27, 29, 30syl3anc 1182 . . . 4  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  ( G " dom  ( F 
\  _I  ) ) 
~~  dom  ( F  \  _I  ) )
3219, 31eqbrtrd 4122 . . 3  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  dom  ( ( ( G  o.  F )  o.  `' G )  \  _I  )  ~~  dom  ( F 
\  _I  ) )
333simp3bi 972 . . . 4  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
3433adantr 451 . . 3  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  dom  ( F  \  _I  )  ~~  2o )
35 entr 6998 . . 3  |-  ( ( dom  ( ( ( G  o.  F )  o.  `' G ) 
\  _I  )  ~~  dom  ( F  \  _I  )  /\  dom  ( F 
\  _I  )  ~~  2o )  ->  dom  (
( ( G  o.  F )  o.  `' G )  \  _I  )  ~~  2o )
3632, 34, 35syl2anc 642 . 2  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  dom  ( ( ( G  o.  F )  o.  `' G )  \  _I  )  ~~  2o )
371, 2pmtrfb 26729 . 2  |-  ( ( ( G  o.  F
)  o.  `' G
)  e.  R  <->  ( D  e.  _V  /\  ( ( G  o.  F )  o.  `' G ) : D -1-1-onto-> D  /\  dom  (
( ( G  o.  F )  o.  `' G )  \  _I  )  ~~  2o ) )
385, 14, 36, 37syl3anbrc 1136 1  |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  (
( G  o.  F
)  o.  `' G
)  e.  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    \ cdif 3225    C_ wss 3228   class class class wbr 4102    _I cid 4383   `'ccnv 4767   dom cdm 4768   ran crn 4769   "cima 4771    o. ccom 4772   -->wf 5330   -1-1->wf1 5331   -1-1-onto->wf1o 5333   ` cfv 5334   2oc2o 6557    ~~ cen 6945  pmTrspcpmtr 26707
This theorem is referenced by:  psgnunilem1  26739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-1o 6563  df-2o 6564  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-pmtr 26708
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