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Theorem gimfn 16877
Description: The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
gimfn  |- GrpIso  Fn  ( Grp  X.  Grp )

Proof of Theorem gimfn
Dummy variables  g 
s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-gim 16875 . 2  |- GrpIso  =  ( s  e.  Grp , 
t  e.  Grp  |->  { g  e.  ( s 
GrpHom  t )  |  g : ( Base `  s
)
-1-1-onto-> ( Base `  t ) } )
2 ovex 6324 . . 3  |-  ( s 
GrpHom  t )  e.  _V
32rabex 4567 . 2  |-  { g  e.  ( s  GrpHom  t )  |  g : ( Base `  s
)
-1-1-onto-> ( Base `  t ) }  e.  _V
41, 3fnmpt2i 6867 1  |- GrpIso  Fn  ( Grp  X.  Grp )
Colors of variables: wff setvar class
Syntax hints:   {crab 2777    X. cxp 4843    Fn wfn 5587   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   Basecbs 15081   Grpcgrp 16621    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-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-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-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-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-gim 16875
This theorem is referenced by:  brgic  16885  gicer  16892
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