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Theorem ghomgsg 29208
Description: A group homomorphism from  G to  H is also a group homomorphism from  G to its image in  H. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomgsg.1  |-  Y  =  ran  F
ghomgsg.2  |-  S  =  ( H  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghomgsg  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )

Proof of Theorem ghomgsg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . 4  |-  ran  G  =  ran  G
2 ghomgsg.1 . . . 4  |-  Y  =  ran  F
3 ghomgsg.2 . . . 4  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 eqid 2457 . . . 4  |-  ran  S  =  ran  S
51, 2, 3, 4ghomfo 29206 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G
-onto->
ran  S )
6 fof 5801 . . 3  |-  ( F : ran  G -onto-> ran  S  ->  F : ran  G --> ran  S )
75, 6syl 16 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G --> ran  S )
8 eqid 2457 . . . . . 6  |-  ran  H  =  ran  H
91, 8elghomOLD 25491 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
109biimp3a 1328 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
1110simprd 463 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )
12 ffvelrn 6030 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  x  e. 
ran  G )  -> 
( F `  x
)  e.  ran  S
)
13 ffvelrn 6030 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  y  e. 
ran  G )  -> 
( F `  y
)  e.  ran  S
)
1412, 13anim12dan 837 . . . . . . 7  |-  ( ( F : ran  G --> ran  S  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( F `  x )  e.  ran  S  /\  ( F `  y )  e.  ran  S ) )
157, 14sylan 471 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( F `
 x )  e. 
ran  S  /\  ( F `  y )  e.  ran  S ) )
162, 3ghomgrp 29205 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
174subgoov 25433 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  H )  /\  (
( F `  x
)  e.  ran  S  /\  ( F `  y
)  e.  ran  S
) )  ->  (
( F `  x
) S ( F `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
1816, 17sylan 471 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( ( F `  x )  e.  ran  S  /\  ( F `  y )  e.  ran  S ) )  ->  ( ( F `
 x ) S ( F `  y
) )  =  ( ( F `  x
) H ( F `
 y ) ) )
1915, 18syldan 470 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( F `
 x ) S ( F `  y
) )  =  ( ( F `  x
) H ( F `
 y ) ) )
2019eqeq1d 2459 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( ( F `  x ) S ( F `  y ) )  =  ( F `  (
x G y ) )  <->  ( ( F `
 x ) H ( F `  y
) )  =  ( F `  ( x G y ) ) ) )
21202ralbidva 2899 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `  (
x G y ) )  <->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
2211, 21mpbird 232 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) )
23 issubgo 25431 . . . . 5  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2416, 23sylib 196 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2524simp2d 1009 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
261, 4elghomOLD 25491 . . . . 5  |-  ( ( G  e.  GrpOp  /\  S  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  S )  <->  ( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
2726biimprd 223 . . . 4  |-  ( ( G  e.  GrpOp  /\  S  e.  GrpOp )  ->  (
( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x
) S ( F `
 y ) )  =  ( F `  ( x G y ) ) )  ->  F  e.  ( G GrpOpHom  S ) ) )
28273adant3 1016 . . 3  |-  ( ( G  e.  GrpOp  /\  S  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  ( G GrpOpHom  S ) ) )
2925, 28syld3an2 1275 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  ( G GrpOpHom  S ) ) )
307, 22, 29mp2and 679 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471    X. cxp 5006   ran crn 5009    |` cres 5010   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296   GrpOpcgr 25314   SubGrpOpcsubgo 25429   GrpOpHom cghomOLD 25485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-grpo 25319  df-gid 25320  df-ginv 25321  df-subgo 25430  df-ghomOLD 25486
This theorem is referenced by: (None)
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