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Theorem trran2 25393
Description: The range of a right translation. The term  A is a constant:  x is not present. (Contributed by FL, 21-Jun-2010.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
Assertion
Ref Expression
trran2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ran  F  =  X )
Distinct variable groups:    x, A    x, G    x, X
Allowed substitution hint:    F( x)

Proof of Theorem trran2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 trfun.2 . . 3  |-  F  =  ( x  e.  X  |->  ( x G A ) )
21rnmpt 4925 . 2  |-  ran  F  =  { y  |  E. x  e.  X  y  =  ( x G A ) }
3 trinv.1 . . . . . . . . . . . 12  |-  X  =  ran  G
43grpocl 20867 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  A  e.  X )  ->  (
x G A )  e.  X )
543exp 1150 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( A  e.  X  ->  (
x G A )  e.  X ) ) )
65com23 72 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( x  e.  X  ->  (
x G A )  e.  X ) ) )
76imp 418 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
x  e.  X  -> 
( x G A )  e.  X ) )
87impcom 419 . . . . . . 7  |-  ( ( x  e.  X  /\  ( G  e.  GrpOp  /\  A  e.  X ) )  -> 
( x G A )  e.  X )
9 eleq1 2343 . . . . . . 7  |-  ( y  =  ( x G A )  ->  (
y  e.  X  <->  ( x G A )  e.  X
) )
108, 9syl5ibrcom 213 . . . . . 6  |-  ( ( x  e.  X  /\  ( G  e.  GrpOp  /\  A  e.  X ) )  -> 
( y  =  ( x G A )  ->  y  e.  X
) )
1110expcom 424 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
x  e.  X  -> 
( y  =  ( x G A )  ->  y  e.  X
) ) )
1211rexlimdv 2666 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( E. x  e.  X  y  =  ( x G A )  ->  y  e.  X ) )
13 simpll 730 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  G  e.  GrpOp
)
14 simpr 447 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  e.  X )
15 eqid 2283 . . . . . . . . 9  |-  ( inv `  G )  =  ( inv `  G )
163, 15grpoinvcl 20893 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( inv `  G
) `  A )  e.  X )
1716adantr 451 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( ( inv `  G ) `  A )  e.  X
)
183grpocl 20867 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X  /\  (
( inv `  G
) `  A )  e.  X )  ->  (
y G ( ( inv `  G ) `
 A ) )  e.  X )
1913, 14, 17, 18syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( y G ( ( inv `  G ) `  A
) )  e.  X
)
20 eqid 2283 . . . . . . . . . . 11  |-  (GId `  G )  =  (GId
`  G )
213, 20, 15grpolinv 20895 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( inv `  G
) `  A ) G A )  =  (GId
`  G ) )
2221adantr 451 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( (
( inv `  G
) `  A ) G A )  =  (GId
`  G ) )
2322eqcomd 2288 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  (GId `  G
)  =  ( ( ( inv `  G
) `  A ) G A ) )
2423oveq2d 5874 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( y G (GId `  G )
)  =  ( y G ( ( ( inv `  G ) `
 A ) G A ) ) )
253, 20grporid 20887 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  (
y G (GId `  G ) )  =  y )
2625eqcomd 2288 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  y  =  ( y G (GId `  G )
) )
2726adantlr 695 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  =  ( y G (GId
`  G ) ) )
28 simplr 731 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  A  e.  X )
293grpoass 20870 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  ( ( inv `  G
) `  A )  e.  X  /\  A  e.  X ) )  -> 
( ( y G ( ( inv `  G
) `  A )
) G A )  =  ( y G ( ( ( inv `  G ) `  A
) G A ) ) )
3013, 14, 17, 28, 29syl13anc 1184 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( (
y G ( ( inv `  G ) `
 A ) ) G A )  =  ( y G ( ( ( inv `  G
) `  A ) G A ) ) )
3124, 27, 303eqtr4d 2325 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  =  ( ( y G ( ( inv `  G
) `  A )
) G A ) )
32 oveq1 5865 . . . . . . . 8  |-  ( x  =  ( y G ( ( inv `  G
) `  A )
)  ->  ( x G A )  =  ( ( y G ( ( inv `  G
) `  A )
) G A ) )
3332eqeq2d 2294 . . . . . . 7  |-  ( x  =  ( y G ( ( inv `  G
) `  A )
)  ->  ( y  =  ( x G A )  <->  y  =  ( ( y G ( ( inv `  G
) `  A )
) G A ) ) )
3433rspcev 2884 . . . . . 6  |-  ( ( ( y G ( ( inv `  G
) `  A )
)  e.  X  /\  y  =  ( (
y G ( ( inv `  G ) `
 A ) ) G A ) )  ->  E. x  e.  X  y  =  ( x G A ) )
3519, 31, 34syl2anc 642 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  E. x  e.  X  y  =  ( x G A ) )
3635ex 423 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
y  e.  X  ->  E. x  e.  X  y  =  ( x G A ) ) )
3712, 36impbid 183 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( E. x  e.  X  y  =  ( x G A )  <->  y  e.  X ) )
3837abbi1dv 2399 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  { y  |  E. x  e.  X  y  =  ( x G A ) }  =  X )
392, 38syl5eq 2327 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ran  F  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  trooo  25394  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-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-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-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860
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