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Theorem gicen 15056
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 15048 . 2  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3629 . . 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 15042 . . . . 5  |-  ( f  e.  ( R GrpIso  S
)  ->  f : B
-1-1-onto-> C )
6 fvex 5734 . . . . . . 7  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2505 . . . . . 6  |-  B  e. 
_V
87f1oen 7120 . . . . 5  |-  ( f : B -1-1-onto-> C  ->  B  ~~  C )
95, 8syl 16 . . . 4  |-  ( f  e.  ( R GrpIso  S
)  ->  B  ~~  C )
109exlimiv 1644 . . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   class class class wbr 4204   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    ~~ cen 7098   Basecbs 13461   GrpIso cgim 15036    ~=ph𝑔 cgic 15037
This theorem is referenced by:  cyggic  16845  sconpi1  24918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-en 7102  df-ghm 14996  df-gim 15038  df-gic 15039
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