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Theorem ghomgrp 27476
Description: The image of a group homomorphism from  G to  H is a subgroup of  H. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomgrp.1  |-  Y  =  ran  F
ghomgrp.2  |-  S  =  ( H  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghomgrp  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )

Proof of Theorem ghomgrp
StepHypRef Expression
1 ghomgrp.2 . . 3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
2 ghomgrp.1 . . . 4  |-  Y  =  ran  F
3 xpid11 5172 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  <->  Y  =  ran  F )
4 reseq2 5216 . . . . 5  |-  ( ( Y  X.  Y )  =  ( ran  F  X.  ran  F )  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
53, 4sylbir 213 . . . 4  |-  ( Y  =  ran  F  -> 
( H  |`  ( Y  X.  Y ) )  =  ( H  |`  ( ran  F  X.  ran  F ) ) )
62, 5ax-mp 5 . . 3  |-  ( H  |`  ( Y  X.  Y
) )  =  ( H  |`  ( ran  F  X.  ran  F ) )
71, 6eqtri 2483 . 2  |-  S  =  ( H  |`  ( ran  F  X.  ran  F
) )
8 id 22 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) ) )
9 eqid 2454 . . 3  |-  { <. <.
x ,  x >. ,  x >. }  =  { <. <. x ,  x >. ,  x >. }
10 eqid 2454 . . 3  |-  (  _I  |`  { x } )  =  (  _I  |`  { x } )
118, 9, 10ghomgrplem 27475 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  (
SubGrpOp `  H ) )
127, 11syl5eqel 2546 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   {csn 3988   <.cop 3994    _I cid 4742    X. cxp 4949   ran crn 4952    |` cres 4953   ` cfv 5529  (class class class)co 6203   GrpOpcgr 23852   SubGrpOpcsubgo 23967   GrpOpHom cghom 24023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-grpo 23857  df-gid 23858  df-ginv 23859  df-subgo 23968  df-ghom 24024
This theorem is referenced by:  ghomfo  27477  ghomgsg  27479
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