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Theorem brgic 15019
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 15010 . 2  |-  ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
2 gimfn 15011 . 2  |- GrpIso  Fn  ( Grp  X.  Grp )
31, 2brwitnlem 6718 1  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    =/= wne 2575   (/)c0 3596   class class class wbr 4180    X. cxp 4843  (class class class)co 6048   Grpcgrp 14648   GrpIso cgim 15007    ~=ph𝑔 cgic 15008
This theorem is referenced by:  brgici  15020  giclcl  15022  gicrcl  15023  gicsym  15024  gictr  15025  gicen  15027  gicsubgen  15028  giccyg  15472  gicabl  27139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-suc 4555  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-1o 6691  df-gim 15009  df-gic 15010
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