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Theorem brgici 15803
Description: Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
brgici  |-  ( F  e.  ( R GrpIso  S
)  ->  R  ~=ph𝑔  S )

Proof of Theorem brgici
StepHypRef Expression
1 ne0i 3648 . 2  |-  ( F  e.  ( R GrpIso  S
)  ->  ( R GrpIso  S )  =/=  (/) )
2 brgic 15802 . 2  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
31, 2sylibr 212 1  |-  ( F  e.  ( R GrpIso  S
)  ->  R  ~=ph𝑔  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    =/= wne 2611   (/)c0 3642   class class class wbr 4297  (class class class)co 6096   GrpIso cgim 15790    ~=ph𝑔 cgic 15791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-1o 6925  df-gim 15792  df-gic 15793
This theorem is referenced by:  gicref  15804  gicsym  15807  gictr  15808  oppggic  15881  cygznlem3  18007  pconpi1  27131  isnumbasgrplem1  29462
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