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Theorem gimf1o 16879
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 16878 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : B -1-1-onto-> C
) )
43simprbi 465 1  |-  ( F  e.  ( R GrpIso  S
)  ->  F : B
-1-1-onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   Basecbs 15081    GrpHom cghm 16832   GrpIso cgim 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-ghm 16833  df-gim 16875
This theorem is referenced by:  subggim  16882  gicen  16893  gicsubgen  16894  giccyg  17475  abliso  28338  gicabl  35711
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