Theorem risci 16141

Description: Determine that two rings are isomorphic.

Assertion

Ref	Expression
risci	|- ((R e. Ring /\ S e. Ring /\ F e. (R RngIso S)) -> R~=rS)

Proof of Theorem risci

Step	Hyp	Ref	Expression
1		risc 16140	. . 3 |- ((R e. Ring /\ S e. Ring) -> (R~=rS <-> E.f f e. (R RngIso S)))
2		eleq1 1957	. . . . 5 |- (f = F -> (f e. (R RngIso S) <-> F e. (R RngIso S)))
3	2	cla4egv 2365	. . . 4 |- (F e. (R RngIso S) -> (F e. (R RngIso S) -> E.f f e. (R RngIso S)))
4	3	pm2.43i 78	. . 3 |- (F e. (R RngIso S) -> E.f f e. (R RngIso S))
5	1, 4	syl5bir 227	. 2 |- ((R e. Ring /\ S e. Ring) -> (F e. (R RngIso S) -> R~=rS))
6	5	3impia 1064	1 |- ((R e. Ring /\ S e. Ring /\ F e. (R RngIso S)) -> R~=rS)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  E.wex 1326   class class class wbr 3338  (class class class)co 4884  Ringcring 9463   RngIso crngiso 16115  ~=rcrisc 16116

This theorem is referenced by:  riscer 16142

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-risc 16137

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