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Theorem isgim2 14828
Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 17556. (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 2358 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2358 . . 3  |-  ( Base `  S )  =  (
Base `  S )
31, 2isgim 14825 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
41, 2ghmf1o 14811 . . 3  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F : ( Base `  R
)
-1-1-onto-> ( Base `  S )  <->  `' F  e.  ( S 
GrpHom  R ) ) )
54pm5.32i 618 . 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 240 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 176    /\ wa 358    e. wcel 1710   `'ccnv 4770   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945   Basecbs 13245    GrpHom cghm 14779   GrpIso cgim 14820
This theorem is referenced by:  gimcnv  14830  gimco  14831  gicref  14834  pi1xfrgim  18660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-mnd 14466  df-grp 14588  df-ghm 14780  df-gim 14822
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