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Theorem isgim2 15015
Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 17752. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
isgim2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  `' F  e.  ( S  GrpHom  R ) ) )

Proof of Theorem isgim2
StepHypRef Expression
1 eqid 2412 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2412 . . 3  |-  ( Base `  S )  =  (
Base `  S )
31, 2isgim 15012 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
41, 2ghmf1o 14998 . . 3  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F : ( Base `  R
)
-1-1-onto-> ( Base `  S )  <->  `' F  e.  ( S 
GrpHom  R ) ) )
54pm5.32i 619 . 2  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F : ( Base `  R
)
-1-1-onto-> ( Base `  S )
)  <->  ( F  e.  ( R  GrpHom  S )  /\  `' F  e.  ( S  GrpHom  R ) ) )
63, 5bitri 241 1  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  `' F  e.  ( S  GrpHom  R ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   `'ccnv 4844   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   Basecbs 13432    GrpHom cghm 14966   GrpIso cgim 15007
This theorem is referenced by:  gimcnv  15017  gimco  15018  gicref  15021  pi1xfrgim  19044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-mnd 14653  df-grp 14775  df-ghm 14967  df-gim 15009
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