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Theorem isrngisom 32975
Description: An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
Assertion
Ref Expression
isrngisom  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( F  e.  ( R RngIsom  S )  <->  ( F  e.  ( R RngHomo  S )  /\  `' F  e.  ( S RngHomo  R ) ) ) )

Proof of Theorem isrngisom
Dummy variables  f 
r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngisom 32967 . . . . 5  |- RngIsom  =  ( r  e.  _V , 
s  e.  _V  |->  { f  e.  ( r RngHomo 
s )  |  `' f  e.  ( s RngHomo  r ) } )
21a1i 11 . . . 4  |-  ( ( R  e.  V  /\  S  e.  W )  -> RngIsom 
=  ( r  e. 
_V ,  s  e. 
_V  |->  { f  e.  ( r RngHomo  s )  |  `' f  e.  ( s RngHomo  r ) } ) )
3 oveq12 6279 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r RngHomo  s )  =  ( R RngHomo  S
) )
43adantl 464 . . . . 5  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( r  =  R  /\  s  =  S ) )  -> 
( r RngHomo  s )  =  ( R RngHomo  S
) )
5 oveq12 6279 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  ( s RngHomo  r )  =  ( S RngHomo  R
) )
65ancoms 451 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( s RngHomo  r )  =  ( S RngHomo  R
) )
76adantl 464 . . . . . 6  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( r  =  R  /\  s  =  S ) )  -> 
( s RngHomo  r )  =  ( S RngHomo  R
) )
87eleq2d 2524 . . . . 5  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( r  =  R  /\  s  =  S ) )  -> 
( `' f  e.  ( s RngHomo  r )  <->  `' f  e.  ( S RngHomo  R ) ) )
94, 8rabeqbidv 3101 . . . 4  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( r  =  R  /\  s  =  S ) )  ->  { f  e.  ( r RngHomo  s )  |  `' f  e.  (
s RngHomo  r ) }  =  { f  e.  ( R RngHomo  S )  |  `' f  e.  ( S RngHomo  R ) } )
10 elex 3115 . . . . 5  |-  ( R  e.  V  ->  R  e.  _V )
1110adantr 463 . . . 4  |-  ( ( R  e.  V  /\  S  e.  W )  ->  R  e.  _V )
12 elex 3115 . . . . 5  |-  ( S  e.  W  ->  S  e.  _V )
1312adantl 464 . . . 4  |-  ( ( R  e.  V  /\  S  e.  W )  ->  S  e.  _V )
14 ovex 6298 . . . . . 6  |-  ( R RngHomo  S )  e.  _V
1514rabex 4588 . . . . 5  |-  { f  e.  ( R RngHomo  S
)  |  `' f  e.  ( S RngHomo  R
) }  e.  _V
1615a1i 11 . . . 4  |-  ( ( R  e.  V  /\  S  e.  W )  ->  { f  e.  ( R RngHomo  S )  |  `' f  e.  ( S RngHomo  R ) }  e.  _V )
172, 9, 11, 13, 16ovmpt2d 6403 . . 3  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R RngIsom  S )  =  { f  e.  ( R RngHomo  S )  |  `' f  e.  ( S RngHomo  R ) } )
1817eleq2d 2524 . 2  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( F  e.  ( R RngIsom  S )  <->  F  e.  { f  e.  ( R RngHomo  S )  |  `' f  e.  ( S RngHomo  R ) } ) )
19 cnveq 5165 . . . 4  |-  ( f  =  F  ->  `' f  =  `' F
)
2019eleq1d 2523 . . 3  |-  ( f  =  F  ->  ( `' f  e.  ( S RngHomo  R )  <->  `' F  e.  ( S RngHomo  R )
) )
2120elrab 3254 . 2  |-  ( F  e.  { f  e.  ( R RngHomo  S )  |  `' f  e.  ( S RngHomo  R ) }  <->  ( F  e.  ( R RngHomo  S )  /\  `' F  e.  ( S RngHomo  R ) ) )
2218, 21syl6bb 261 1  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( F  e.  ( R RngIsom  S )  <->  ( F  e.  ( R RngHomo  S )  /\  `' F  e.  ( S RngHomo  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106   `'ccnv 4987  (class class class)co 6270    |-> cmpt2 6272   RngHomo crngh 32964   RngIsom crngs 32965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-rngisom 32967
This theorem is referenced by:  isrngim  32983  rngcinv  33062  rngcinvALTV  33074
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