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Theorem ghomfo 28356
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 2460 . . . . . 6  |-  ran  H  =  ran  H
31, 2elghom 24891 . . . . 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 1323 . . . 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 5722 . . 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 5195 . . . . 5  |-  dom  S  =  dom  ( H  |`  ( Y  X.  Y
) )
10 ghomfo.2 . . . . . . . . 9  |-  Y  =  ran  F
1110, 8ghomgrp 28355 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
12 issubgo 24831 . . . . . . . 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 1004 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
15 ghomfo.4 . . . . . . 7  |-  Z  =  ran  S
1615grpofo 24727 . . . . . 6  |-  ( S  e.  GrpOp  ->  S :
( Z  X.  Z
) -onto-> Z )
17 fof 5786 . . . . . 6  |-  ( S : ( Z  X.  Z ) -onto-> Z  ->  S : ( Z  X.  Z ) --> Z )
18 fdm 5726 . . . . . 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 5728 . . . . . . . . 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 3541 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Y  C_  ran  H )
23 xpss12 5099 . . . . . . 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 5286 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  H  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
262grpofo 24727 . . . . . . . . . 10  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
27 fof 5786 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
28 fdm 5726 . . . . . . . . . 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 3525 . . . . . . . 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 1013 . . . . . 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 2525 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Z  X.  Z )  =  ( Y  X.  Y ) )
35 xpid11 5215 . . . 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 2518 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  =  Z )
38 df-fo 5585 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469    X. cxp 4990   dom cdm 4992   ran crn 4993    |` cres 4994    Fn wfn 5574   -->wf 5575   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24714   SubGrpOpcsubgo 24829   GrpOpHom cghom 24885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-grpo 24719  df-gid 24720  df-ginv 24721  df-subgo 24830  df-ghom 24886
This theorem is referenced by:  ghomcl  28357  ghomgsg  28358  ghomf1olem  28359
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