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Theorem ghmrn 14696
Description: The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmrn  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T )
)

Proof of Theorem ghmrn
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2283 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
31, 2ghmf 14687 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frn 5395 . . 3  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
53, 4syl 15 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  C_  ( Base `  T )
)
6 fdm 5393 . . . . 5  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
73, 6syl 15 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =  ( Base `  S
) )
8 ghmgrp1 14685 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
91grpbn0 14511 . . . . 5  |-  ( S  e.  Grp  ->  ( Base `  S )  =/=  (/) )
108, 9syl 15 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  ( Base `  S )  =/=  (/) )
117, 10eqnetrd 2464 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  dom  F  =/=  (/) )
12 dm0rn0 4895 . . . 4  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2478 . . 3  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 188 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  =/=  (/) )
15 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  S )  =  ( +g  `  S )
16 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  T )  =  ( +g  `  T )
171, 15, 16ghmlin 14688 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( F `  ( c ( +g  `  S ) a ) )  =  ( ( F `  c ) ( +g  `  T
) ( F `  a ) ) )
18 ffn 5389 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
193, 18syl 15 . . . . . . . . . . 11  |-  ( F  e.  ( S  GrpHom  T )  ->  F  Fn  ( Base `  S )
)
20193ad2ant1 976 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  F  Fn  ( Base `  S )
)
211, 15grpcl 14495 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  c  e.  ( Base `  S )  /\  a  e.  ( Base `  S
) )  ->  (
c ( +g  `  S
) a )  e.  ( Base `  S
) )
228, 21syl3an1 1215 . . . . . . . . . 10  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( c
( +g  `  S ) a )  e.  (
Base `  S )
)
23 fnfvelrn 5662 . . . . . . . . . 10  |-  ( ( F  Fn  ( Base `  S )  /\  (
c ( +g  `  S
) a )  e.  ( Base `  S
) )  ->  ( F `  ( c
( +g  `  S ) a ) )  e. 
ran  F )
2420, 22, 23syl2anc 642 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( F `  ( c ( +g  `  S ) a ) )  e.  ran  F
)
2517, 24eqeltrrd 2358 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
)  /\  a  e.  ( Base `  S )
)  ->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F )
26253expia 1153 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
a  e.  ( Base `  S )  ->  (
( F `  c
) ( +g  `  T
) ( F `  a ) )  e. 
ran  F ) )
2726ralrimiv 2625 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  A. a  e.  ( Base `  S
) ( ( F `
 c ) ( +g  `  T ) ( F `  a
) )  e.  ran  F )
28 oveq2 5866 . . . . . . . . . 10  |-  ( b  =  ( F `  a )  ->  (
( F `  c
) ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) ( F `
 a ) ) )
2928eleq1d 2349 . . . . . . . . 9  |-  ( b  =  ( F `  a )  ->  (
( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3029ralrn 5668 . . . . . . . 8  |-  ( F  Fn  ( Base `  S
)  ->  ( A. b  e.  ran  F ( ( F `  c
) ( +g  `  T
) b )  e. 
ran  F  <->  A. a  e.  (
Base `  S )
( ( F `  c ) ( +g  `  T ) ( F `
 a ) )  e.  ran  F ) )
3119, 30syl 15 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  ( A. b  e.  ran  F ( ( F `  c
) ( +g  `  T
) b )  e. 
ran  F  <->  A. a  e.  (
Base `  S )
( ( F `  c ) ( +g  `  T ) ( F `
 a ) )  e.  ran  F ) )
3231adantr 451 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  <->  A. a  e.  ( Base `  S
) ( ( F `
 c ) ( +g  `  T ) ( F `  a
) )  e.  ran  F ) )
3327, 32mpbird 223 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F )
34 eqid 2283 . . . . . . 7  |-  ( inv g `  S )  =  ( inv g `  S )
35 eqid 2283 . . . . . . 7  |-  ( inv g `  T )  =  ( inv g `  T )
361, 34, 35ghminv 14690 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 c ) )  =  ( ( inv g `  T ) `
 ( F `  c ) ) )
3719adantr 451 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  F  Fn  ( Base `  S
) )
381, 34grpinvcl 14527 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  c  e.  ( Base `  S ) )  -> 
( ( inv g `  S ) `  c
)  e.  ( Base `  S ) )
398, 38sylan 457 . . . . . . 7  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
( inv g `  S ) `  c
)  e.  ( Base `  S ) )
40 fnfvelrn 5662 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  (
( inv g `  S ) `  c
)  e.  ( Base `  S ) )  -> 
( F `  (
( inv g `  S ) `  c
) )  e.  ran  F )
4137, 39, 40syl2anc 642 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 c ) )  e.  ran  F )
4236, 41eqeltrrd 2358 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
)
4333, 42jca 518 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  c  e.  ( Base `  S
) )  ->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T ) b )  e.  ran  F  /\  ( ( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) )
4443ralrimiva 2626 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. c  e.  ( Base `  S
) ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) )
45 oveq1 5865 . . . . . . . 8  |-  ( a  =  ( F `  c )  ->  (
a ( +g  `  T
) b )  =  ( ( F `  c ) ( +g  `  T ) b ) )
4645eleq1d 2349 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( a ( +g  `  T ) b )  e.  ran  F  <->  ( ( F `  c )
( +g  `  T ) b )  e.  ran  F ) )
4746ralbidv 2563 . . . . . 6  |-  ( a  =  ( F `  c )  ->  ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  <->  A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F ) )
48 fveq2 5525 . . . . . . 7  |-  ( a  =  ( F `  c )  ->  (
( inv g `  T ) `  a
)  =  ( ( inv g `  T
) `  ( F `  c ) ) )
4948eleq1d 2349 . . . . . 6  |-  ( a  =  ( F `  c )  ->  (
( ( inv g `  T ) `  a
)  e.  ran  F  <->  ( ( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) )
5047, 49anbi12d 691 . . . . 5  |-  ( a  =  ( F `  c )  ->  (
( A. b  e. 
ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
)  <->  ( A. b  e.  ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5150ralrn 5668 . . . 4  |-  ( F  Fn  ( Base `  S
)  ->  ( A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  /\  ( ( inv g `  T ) `  a
)  e.  ran  F
)  <->  A. c  e.  (
Base `  S )
( A. b  e. 
ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5219, 51syl 15 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T ) b )  e.  ran  F  /\  ( ( inv g `  T ) `  a
)  e.  ran  F
)  <->  A. c  e.  (
Base `  S )
( A. b  e. 
ran  F ( ( F `  c ) ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  ( F `  c )
)  e.  ran  F
) ) )
5344, 52mpbird 223 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  A. a  e.  ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
) )
54 ghmgrp2 14686 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
552, 16, 35issubg2 14636 . . 3  |-  ( T  e.  Grp  ->  ( ran  F  e.  (SubGrp `  T )  <->  ( ran  F 
C_  ( Base `  T
)  /\  ran  F  =/=  (/)  /\  A. a  e. 
ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
) ) ) )
5654, 55syl 15 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( ran  F  e.  (SubGrp `  T
)  <->  ( ran  F  C_  ( Base `  T
)  /\  ran  F  =/=  (/)  /\  A. a  e. 
ran  F ( A. b  e.  ran  F ( a ( +g  `  T
) b )  e. 
ran  F  /\  (
( inv g `  T ) `  a
)  e.  ran  F
) ) ) )
575, 14, 53, 56mpbir3and 1135 1  |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363  SubGrpcsubg 14615    GrpHom cghm 14680
This theorem is referenced by:  ghmima  14703  cayley  14789
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-ghm 14681
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