Theorem risci 33786

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.

Step	Hyp	Ref	Expression
1		elex2 3216	. . 3 |- ( F e. ( R RngIso S ) -> E. f f e. ( R RngIso S ) )
2		risc 33785	. . 3 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( R ~=R S <-> E. f f e. ( R RngIso S ) ) )
3	1, 2	syl5ibr 236	. 2 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngIso S ) -> R ~=R S ) )
4	3	3impia 1261	1 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngIso S ) ) -> R ~=R S )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RingOpscrngo 33693   RngIso crngiso 33760   ~=R crisc 33761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-risc 33782

This theorem is referenced by:  riscer  33787