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Theorem risci 28933
Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
risci  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S
)

Proof of Theorem risci
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 elex2 3082 . . 3  |-  ( F  e.  ( R  RngIso  S )  ->  E. f 
f  e.  ( R 
RngIso  S ) )
2 risc 28932 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=r  S  <->  E. f 
f  e.  ( R 
RngIso  S ) ) )
31, 2syl5ibr 221 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  R  ~=r 
S ) )
433impia 1185 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   E.wex 1587    e. wcel 1758   class class class wbr 4392  (class class class)co 6192   RingOpscrngo 23999    RngIso crngiso 28907    ~=r crisc 28908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-iota 5481  df-fv 5526  df-ov 6195  df-risc 28929
This theorem is referenced by:  riscer  28934
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