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Theorem gicen 14991
Description: Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
gicen.b  |-  B  =  ( Base `  R
)
gicen.c  |-  C  =  ( Base `  S
)
Assertion
Ref Expression
gicen  |-  ( R 
~=ph𝑔  S  ->  B  ~~  C
)

Proof of Theorem gicen
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brgic 14983 . 2  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3580 . . 3  |-  ( ( R GrpIso  S )  =/=  (/) 
<->  E. f  f  e.  ( R GrpIso  S ) )
3 gicen.b . . . . . 6  |-  B  =  ( Base `  R
)
4 gicen.c . . . . . 6  |-  C  =  ( Base `  S
)
53, 4gimf1o 14977 . . . . 5  |-  ( f  e.  ( R GrpIso  S
)  ->  f : B
-1-1-onto-> C )
6 fvex 5682 . . . . . . 7  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2457 . . . . . 6  |-  B  e. 
_V
87f1oen 7064 . . . . 5  |-  ( f : B -1-1-onto-> C  ->  B  ~~  C )
95, 8syl 16 . . . 4  |-  ( f  e.  ( R GrpIso  S
)  ->  B  ~~  C )
109exlimiv 1641 . . 3  |-  ( E. f  f  e.  ( R GrpIso  S )  ->  B  ~~  C )
112, 10sylbi 188 . 2  |-  ( ( R GrpIso  S )  =/=  (/)  ->  B  ~~  C
)
121, 11sylbi 188 1  |-  ( R 
~=ph𝑔  S  ->  B  ~~  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899   (/)c0 3571   class class class wbr 4153   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    ~~ cen 7042   Basecbs 13396   GrpIso cgim 14971    ~=ph𝑔 cgic 14972
This theorem is referenced by:  cyggic  16776  sconpi1  24705
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-1o 6660  df-en 7046  df-ghm 14931  df-gim 14973  df-gic 14974
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