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Theorem ghomgsg 25057
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 2404 . . . 4  |-  ran  G  =  ran  G
2 ghomgsg.1 . . . 4  |-  Y  =  ran  F
3 ghomgsg.2 . . . 4  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 eqid 2404 . . . 4  |-  ran  S  =  ran  S
51, 2, 3, 4ghomfo 25055 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G
-onto->
ran  S )
6 fof 5612 . . 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 2404 . . . . . 6  |-  ran  H  =  ran  H
91, 8elghom 21904 . . . . 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 1283 . . . 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 450 . . 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 5827 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  x  e. 
ran  G )  -> 
( F `  x
)  e.  ran  S
)
13 ffvelrn 5827 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  y  e. 
ran  G )  -> 
( F `  y
)  e.  ran  S
)
1412, 13anim12dan 811 . . . . . . 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 458 . . . . . 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 25054 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
174subgoov 21846 . . . . . . 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 458 . . . . . 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 457 . . . . 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 2412 . . . 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 2706 . . 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 224 . 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 21844 . . . . 5  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2416, 23sylib 189 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2524simp2d 970 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
261, 4elghom 21904 . . . . 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 215 . . . 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 977 . . 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 1231 . 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 661 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280    X. cxp 4835   ran crn 4838    |` cres 4839   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040   GrpOpcgr 21727   SubGrpOpcsubgo 21842   GrpOpHom cghom 21898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-grpo 21732  df-gid 21733  df-ginv 21734  df-subgo 21843  df-ghom 21899
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