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Theorem ghomgsg 24587
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 2366 . . . 4  |-  ran  G  =  ran  G
2 ghomgsg.1 . . . 4  |-  Y  =  ran  F
3 ghomgsg.2 . . . 4  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 eqid 2366 . . . 4  |-  ran  S  =  ran  S
51, 2, 3, 4ghomfo 24585 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G
-onto->
ran  S )
6 fof 5557 . . 3  |-  ( F : ran  G -onto-> ran  S  ->  F : ran  G --> ran  S )
75, 6syl 15 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G --> ran  S )
8 eqid 2366 . . . . . 6  |-  ran  H  =  ran  H
91, 8elghom 21341 . . . . 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 1282 . . . 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 449 . . 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 5770 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  x  e. 
ran  G )  -> 
( F `  x
)  e.  ran  S
)
13 ffvelrn 5770 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  y  e. 
ran  G )  -> 
( F `  y
)  e.  ran  S
)
1412, 13anim12dan 810 . . . . . . 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 457 . . . . . 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 24584 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
174subgoov 21283 . . . . . . 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 457 . . . . . 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 456 . . . . 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 2374 . . . 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 2668 . . 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 223 . 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 21281 . . . . 5  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2416, 23sylib 188 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2524simp2d 969 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
261, 4elghom 21341 . . . . 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 214 . . . 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 976 . . 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 1230 . 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 660 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628    C_ wss 3238    X. cxp 4790   ran crn 4793    |` cres 4794   -->wf 5354   -onto->wfo 5356   ` cfv 5358  (class class class)co 5981   GrpOpcgr 21164   SubGrpOpcsubgo 21279   GrpOpHom cghom 21335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-grpo 21169  df-gid 21170  df-ginv 21171  df-subgo 21280  df-ghom 21336
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