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Theorem gimf1o 16125
Description: An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
isgim.b  |-  B  =  ( Base `  R
)
isgim.c  |-  C  =  ( Base `  S
)
Assertion
Ref Expression
gimf1o  |-  ( F  e.  ( R GrpIso  S
)  ->  F : B
-1-1-onto-> C )

Proof of Theorem gimf1o
StepHypRef Expression
1 isgim.b . . 3  |-  B  =  ( Base `  R
)
2 isgim.c . . 3  |-  C  =  ( Base `  S
)
31, 2isgim 16124 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C
) )
43simprbi 464 1  |-  ( F  e.  ( R GrpIso  S
)  ->  F : B
-1-1-onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285   Basecbs 14493    GrpHom cghm 16078   GrpIso cgim 16119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-ghm 16079  df-gim 16121
This theorem is referenced by:  subggim  16128  gicen  16139  gicsubgen  16140  giccyg  16717  abliso  27445  gicabl  30878
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