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Theorem ghomfo 23998
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 2283 . . . . . 6  |-  ran  H  =  ran  H
31, 2elghom 21030 . . . . 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 1281 . . . 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 445 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
6 ffn 5389 . . 3  |-  ( F : X --> ran  H  ->  F  Fn  X )
75, 6syl 15 . 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 4880 . . . . 5  |-  dom  S  =  dom  ( H  |`  ( Y  X.  Y
) )
10 ghomfo.2 . . . . . . . . 9  |-  Y  =  ran  F
1110, 8ghomgrp 23997 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
12 issubgo 20970 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1311, 12sylib 188 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1413simp2d 968 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
15 ghomfo.4 . . . . . . . 8  |-  Z  =  ran  S
1615grpofo 20866 . . . . . . 7  |-  ( S  e.  GrpOp  ->  S :
( Z  X.  Z
) -onto-> Z )
17 fof 5451 . . . . . . 7  |-  ( S : ( Z  X.  Z ) -onto-> Z  ->  S : ( Z  X.  Z ) --> Z )
18 fdm 5393 . . . . . . 7  |-  ( S : ( Z  X.  Z ) --> Z  ->  dom  S  =  ( Z  X.  Z ) )
1916, 17, 183syl 18 . . . . . 6  |-  ( S  e.  GrpOp  ->  dom  S  =  ( Z  X.  Z
) )
2014, 19syl 15 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  S  =  ( Z  X.  Z
) )
21 frn 5395 . . . . . . . . 9  |-  ( F : X --> ran  H  ->  ran  F  C_  ran  H )
225, 21syl 15 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  C_  ran  H )
2310, 22syl5eqss 3222 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Y  C_  ran  H )
24 xpss12 4792 . . . . . . 7  |-  ( ( Y  C_  ran  H  /\  Y  C_  ran  H )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
2523, 23, 24syl2anc 642 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
26 ssdmres 4977 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  H  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
272grpofo 20866 . . . . . . . . . 10  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
28 fof 5451 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
29 fdm 5393 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) --> ran 
H  ->  dom  H  =  ( ran  H  X.  ran  H ) )
3027, 28, 293syl 18 . . . . . . . . 9  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
3130sseq2d 3206 . . . . . . . 8  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  dom  H  <->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) ) )
3226, 31syl5rbbr 251 . . . . . . 7  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  ( ran  H  X.  ran  H )  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) ) )
33323ad2ant2 977 . . . . . 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 ) ) )
3425, 33mpbid 201 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
359, 20, 343eqtr3a 2339 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Z  X.  Z )  =  ( Y  X.  Y ) )
36 xpid11 4900 . . . 4  |-  ( ( Z  X.  Z )  =  ( Y  X.  Y )  <->  Z  =  Y )
3735, 36sylib 188 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Z  =  Y )
3837, 10syl6req 2332 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  =  Z )
39 df-fo 5261 . 2  |-  ( F : X -onto-> Z  <->  ( F  Fn  X  /\  ran  F  =  Z ) )
407, 38, 39sylanbrc 645 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   SubGrpOpcsubgo 20968   GrpOpHom cghom 21024
This theorem is referenced by:  ghomcl  23999  ghomgsg  24000  ghomf1olem  24001
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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