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Theorem ghomfo 30311
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 2423 . . . . . 6  |-  ran  H  =  ran  H
31, 2elghomOLD 26083 . . . . 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 1365 . . . 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 461 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
6 ffn 5744 . . 3  |-  ( F : X --> ran  H  ->  F  Fn  X )
75, 6syl 17 . 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 5053 . . . . 5  |-  dom  S  =  dom  ( H  |`  ( Y  X.  Y
) )
10 ghomfo.2 . . . . . . . . 9  |-  Y  =  ran  F
1110, 8ghomgrp 30310 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
12 issubgo 26023 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1311, 12sylib 200 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1413simp2d 1019 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
15 ghomfo.4 . . . . . . 7  |-  Z  =  ran  S
1615grpofo 25919 . . . . . 6  |-  ( S  e.  GrpOp  ->  S :
( Z  X.  Z
) -onto-> Z )
17 fof 5808 . . . . . 6  |-  ( S : ( Z  X.  Z ) -onto-> Z  ->  S : ( Z  X.  Z ) --> Z )
18 fdm 5748 . . . . . 6  |-  ( S : ( Z  X.  Z ) --> Z  ->  dom  S  =  ( Z  X.  Z ) )
1914, 16, 17, 184syl 19 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  S  =  ( Z  X.  Z
) )
20 frn 5750 . . . . . . . . 9  |-  ( F : X --> ran  H  ->  ran  F  C_  ran  H )
215, 20syl 17 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  C_  ran  H )
2210, 21syl5eqss 3509 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Y  C_  ran  H )
23 xpss12 4957 . . . . . . 7  |-  ( ( Y  C_  ran  H  /\  Y  C_  ran  H )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
2422, 22, 23syl2anc 666 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
25 ssdmres 5143 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  H  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
262grpofo 25919 . . . . . . . . . 10  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
27 fof 5808 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
28 fdm 5748 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) --> ran 
H  ->  dom  H  =  ( ran  H  X.  ran  H ) )
2926, 27, 283syl 18 . . . . . . . . 9  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
3029sseq2d 3493 . . . . . . . 8  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  dom  H  <->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) ) )
3125, 30syl5rbbr 264 . . . . . . 7  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  ( ran  H  X.  ran  H )  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) ) )
32313ad2ant2 1028 . . . . . 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 214 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
349, 19, 333eqtr3a 2488 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Z  X.  Z )  =  ( Y  X.  Y ) )
35 xpid11 5073 . . . 4  |-  ( ( Z  X.  Z )  =  ( Y  X.  Y )  <->  Z  =  Y )
3634, 35sylib 200 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Z  =  Y )
3736, 10syl6req 2481 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  =  Z )
38 df-fo 5605 . 2  |-  ( F : X -onto-> Z  <->  ( F  Fn  X  /\  ran  F  =  Z ) )
397, 37, 38sylanbrc 669 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776    C_ wss 3437    X. cxp 4849   dom cdm 4851   ran crn 4852    |` cres 4853    Fn wfn 5594   -->wf 5595   -onto->wfo 5597   ` cfv 5599  (class class class)co 6303   GrpOpcgr 25906   SubGrpOpcsubgo 26021   GrpOpHom cghomOLD 26077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-grpo 25911  df-gid 25912  df-ginv 25913  df-subgo 26022  df-ghomOLD 26078
This theorem is referenced by:  ghomcl  30312  ghomgsg  30313  ghomf1olem  30314
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