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Theorem brgic 14943
Description: The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
brgic  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )

Proof of Theorem brgic
StepHypRef Expression
1 df-gic 14934 . 2  |-  ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 gimfn 14935 . 2  |- GrpIso  Fn  ( Grp  X.  Grp )
31, 2brwitnlem 6648 1  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    =/= wne 2529   (/)c0 3543   class class class wbr 4125    X. cxp 4790  (class class class)co 5981   Grpcgrp 14572   GrpIso cgim 14931    ~=ph𝑔 cgic 14932
This theorem is referenced by:  brgici  14944  giclcl  14946  gicrcl  14947  gicsym  14948  gictr  14949  gicen  14951  gicsubgen  14952  giccyg  15396  gicabl  26854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-suc 4501  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-1o 6621  df-gim 14933  df-gic 14934
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