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Theorem prsubrtr 25399
Description: The product of a subset  B of  X by an element of  X is the image of  B by a right translation. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
prsubrtr.1  |-  H  =  ( cset `  G
)
Assertion
Ref Expression
prsubrtr  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( B H { A } )  =  ( F " B ) )
Distinct variable groups:    x, A    x, B    x, G    x, X
Allowed substitution hints:    F( x)    H( x)

Proof of Theorem prsubrtr
Dummy variables  a 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . . . 7  |-  ( y  =  A  ->  (
x G y )  =  ( x G A ) )
21eqeq2d 2294 . . . . . 6  |-  ( y  =  A  ->  (
a  =  ( x G y )  <->  a  =  ( x G A ) ) )
32rexsng 3673 . . . . 5  |-  ( A  e.  X  ->  ( E. y  e.  { A } a  =  ( x G y )  <-> 
a  =  ( x G A ) ) )
43rexbidv 2564 . . . 4  |-  ( A  e.  X  ->  ( E. x  e.  B  E. y  e.  { A } a  =  ( x G y )  <->  E. x  e.  B  a  =  ( x G A ) ) )
543ad2ant2 977 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( E. x  e.  B  E. y  e. 
{ A } a  =  ( x G y )  <->  E. x  e.  B  a  =  ( x G A ) ) )
6 simp1 955 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  G  e.  GrpOp )
7 trinv.1 . . . . . . . . 9  |-  X  =  ran  G
8 grporndm 20877 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
97, 8syl5eq 2327 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  =  dom  dom  G )
109pweqd 3630 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P X  =  ~P dom  dom  G
)
1110eleq2d 2350 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( B  e.  ~P X  <->  B  e.  ~P dom  dom  G )
)
1211biimpa 470 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  ~P X )  ->  B  e.  ~P dom  dom 
G )
13123adant2 974 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  B  e.  ~P dom  dom 
G )
149eleq2d 2350 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
1514biimpa 470 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A  e.  dom  dom  G )
16153adant3 975 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  A  e.  dom  dom  G
)
17 snelpwi 4220 . . . . 5  |-  ( A  e.  dom  dom  G  ->  { A }  e.  ~P dom  dom  G )
1816, 17syl 15 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  { A }  e.  ~P dom  dom  G )
19 eqid 2283 . . . . 5  |-  dom  dom  G  =  dom  dom  G
20 prsubrtr.1 . . . . 5  |-  H  =  ( cset `  G
)
2119, 20iscst3 25176 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  ~P dom  dom  G  /\  { A }  e.  ~P dom  dom  G )  ->  ( a  e.  ( B H { A } )  <->  E. x  e.  B  E. y  e.  { A } a  =  ( x G y ) ) )
226, 13, 18, 21syl3anc 1182 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( B H { A } )  <->  E. x  e.  B  E. y  e.  { A } a  =  ( x G y ) ) )
23 df-ima 4702 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
24 trfun.2 . . . . . . . . 9  |-  F  =  ( x  e.  X  |->  ( x G A ) )
2524reseq1i 4951 . . . . . . . 8  |-  ( F  |`  B )  =  ( ( x  e.  X  |->  ( x G A ) )  |`  B )
26 elpwi 3633 . . . . . . . . . 10  |-  ( B  e.  ~P X  ->  B  C_  X )
27263ad2ant3 978 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  B  C_  X )
28 resmpt 5000 . . . . . . . . 9  |-  ( B 
C_  X  ->  (
( x  e.  X  |->  ( x G A ) )  |`  B )  =  ( x  e.  B  |->  ( x G A ) ) )
2927, 28syl 15 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( ( x  e.  X  |->  ( x G A ) )  |`  B )  =  ( x  e.  B  |->  ( x G A ) ) )
3025, 29syl5eq 2327 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F  |`  B )  =  ( x  e.  B  |->  ( x G A ) ) )
3130rneqd 4906 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  ran  ( F  |`  B )  =  ran  ( x  e.  B  |->  ( x G A ) ) )
3223, 31syl5eq 2327 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F " B
)  =  ran  (
x  e.  B  |->  ( x G A ) ) )
3332eleq2d 2350 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( F " B )  <-> 
a  e.  ran  (
x  e.  B  |->  ( x G A ) ) ) )
34 eqid 2283 . . . . 5  |-  ( x  e.  B  |->  ( x G A ) )  =  ( x  e.  B  |->  ( x G A ) )
35 ovex 5883 . . . . 5  |-  ( x G A )  e. 
_V
3634, 35elrnmpti 4930 . . . 4  |-  ( a  e.  ran  ( x  e.  B  |->  ( x G A ) )  <->  E. x  e.  B  a  =  ( x G A ) )
3733, 36syl6bb 252 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( F " B )  <->  E. x  e.  B  a  =  ( x G A ) ) )
385, 22, 373bitr4d 276 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( B H { A } )  <->  a  e.  ( F " B ) ) )
3938eqrdv 2281 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( B H { A } )  =  ( F " B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   {csn 3640    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   csetccst 25172
This theorem is referenced by:  caytr  25400
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-grpo 20858  df-cst 25173
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