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Theorem ghomgsg 24000
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 2283 . . . 4  |-  ran  G  =  ran  G
2 ghomgsg.1 . . . 4  |-  Y  =  ran  F
3 ghomgsg.2 . . . 4  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 eqid 2283 . . . 4  |-  ran  S  =  ran  S
51, 2, 3, 4ghomfo 23998 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G
-onto->
ran  S )
6 fof 5451 . . 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 2283 . . . . . 6  |-  ran  H  =  ran  H
91, 8elghom 21030 . . . . 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 1281 . . . 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 5663 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  x  e. 
ran  G )  -> 
( F `  x
)  e.  ran  S
)
13 ffvelrn 5663 . . . . . . . 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 23997 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
174subgoov 20972 . . . . . . 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 2291 . . . 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 2583 . . 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 20970 . . . . 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 968 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
261, 4elghom 21030 . . . . 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 975 . . 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 1229 . 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 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    X. cxp 4687   ran crn 4690    |` cres 4691   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   SubGrpOpcsubgo 20968   GrpOpHom cghom 21024
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-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-subgo 20969  df-ghom 21025
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