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Theorem risci 26594
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 2960 . . 3  |-  ( F  e.  ( R  RngIso  S )  ->  E. f 
f  e.  ( R 
RngIso  S ) )
2 risc 26593 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=r  S  <->  E. f 
f  e.  ( R 
RngIso  S ) ) )
31, 2syl5ibr 213 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  R  ~=r 
S ) )
433impia 1150 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   RingOpscrngo 21955    RngIso crngiso 26568    ~=r crisc 26569
This theorem is referenced by:  riscer  26595
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-iota 5410  df-fv 5454  df-ov 6076  df-risc 26590
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