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Theorem ghomfo 27446
Description: A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomfo.1  |-  X  =  ran  G
ghomfo.2  |-  Y  =  ran  F
ghomfo.3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
ghomfo.4  |-  Z  =  ran  S
Assertion
Ref Expression
ghomfo  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )

Proof of Theorem ghomfo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomfo.1 . . . . . 6  |-  X  =  ran  G
2 eqid 2451 . . . . . 6  |-  ran  H  =  ran  H
31, 2elghom 23987 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> ran  H  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) ) ) )
43biimp3a 1319 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : X
--> ran  H  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) ) )
54simpld 459 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
6 ffn 5659 . . 3  |-  ( F : X --> ran  H  ->  F  Fn  X )
75, 6syl 16 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  Fn  X
)
8 ghomfo.3 . . . . . 6  |-  S  =  ( H  |`  ( Y  X.  Y ) )
98dmeqi 5141 . . . . 5  |-  dom  S  =  dom  ( H  |`  ( Y  X.  Y
) )
10 ghomfo.2 . . . . . . . . 9  |-  Y  =  ran  F
1110, 8ghomgrp 27445 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
12 issubgo 23927 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1311, 12sylib 196 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1413simp2d 1001 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
15 ghomfo.4 . . . . . . 7  |-  Z  =  ran  S
1615grpofo 23823 . . . . . 6  |-  ( S  e.  GrpOp  ->  S :
( Z  X.  Z
) -onto-> Z )
17 fof 5720 . . . . . 6  |-  ( S : ( Z  X.  Z ) -onto-> Z  ->  S : ( Z  X.  Z ) --> Z )
18 fdm 5663 . . . . . 6  |-  ( S : ( Z  X.  Z ) --> Z  ->  dom  S  =  ( Z  X.  Z ) )
1914, 16, 17, 184syl 21 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  S  =  ( Z  X.  Z
) )
20 frn 5665 . . . . . . . . 9  |-  ( F : X --> ran  H  ->  ran  F  C_  ran  H )
215, 20syl 16 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  C_  ran  H )
2210, 21syl5eqss 3500 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Y  C_  ran  H )
23 xpss12 5045 . . . . . . 7  |-  ( ( Y  C_  ran  H  /\  Y  C_  ran  H )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
2422, 22, 23syl2anc 661 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
25 ssdmres 5232 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  H  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
262grpofo 23823 . . . . . . . . . 10  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
27 fof 5720 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
28 fdm 5663 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) --> ran 
H  ->  dom  H  =  ( ran  H  X.  ran  H ) )
2926, 27, 283syl 20 . . . . . . . . 9  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
3029sseq2d 3484 . . . . . . . 8  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  dom  H  <->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) ) )
3125, 30syl5rbbr 260 . . . . . . 7  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  ( ran  H  X.  ran  H )  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) ) )
32313ad2ant2 1010 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( Y  X.  Y )  C_  ( ran  H  X.  ran  H )  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) ) )
3324, 32mpbid 210 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
349, 19, 333eqtr3a 2516 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Z  X.  Z )  =  ( Y  X.  Y ) )
35 xpid11 5161 . . . 4  |-  ( ( Z  X.  Z )  =  ( Y  X.  Y )  <->  Z  =  Y )
3634, 35sylib 196 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Z  =  Y )
3736, 10syl6req 2509 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  =  Z )
38 df-fo 5524 . 2  |-  ( F : X -onto-> Z  <->  ( F  Fn  X  /\  ran  F  =  Z ) )
397, 37, 38sylanbrc 664 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3428    X. cxp 4938   dom cdm 4940   ran crn 4941    |` cres 4942    Fn wfn 5513   -->wf 5514   -onto->wfo 5516   ` cfv 5518  (class class class)co 6192   GrpOpcgr 23810   SubGrpOpcsubgo 23925   GrpOpHom cghom 23981
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-grpo 23815  df-gid 23816  df-ginv 23817  df-subgo 23926  df-ghom 23982
This theorem is referenced by:  ghomcl  27447  ghomgsg  27448  ghomf1olem  27449
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