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Theorem islmim 16139
Description: An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
islmim.b  |-  B  =  ( Base `  R
)
islmim.c  |-  C  =  ( Base `  S
)
Assertion
Ref Expression
islmim  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )

Proof of Theorem islmim
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmim 16104 . . 3  |- LMIso  =  ( a  e.  LMod ,  b  e.  LMod  |->  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
2 ovex 6109 . . . 4  |-  ( a LMHom 
b )  e.  _V
32rabex 4357 . . 3  |-  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
4 oveq12 6093 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a LMHom  b )  =  ( R LMHom  S
) )
5 fveq2 5731 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
6 islmim.b . . . . . 6  |-  B  =  ( Base `  R
)
75, 6syl6eqr 2488 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
8 fveq2 5731 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
9 islmim.c . . . . . 6  |-  C  =  ( Base `  S
)
108, 9syl6eqr 2488 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
11 f1oeq23 5671 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
127, 10, 11syl2an 465 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
134, 12rabeqbidv 2953 . . 3  |-  ( ( a  =  R  /\  b  =  S )  ->  { c  e.  ( a LMHom  b )  |  c : ( Base `  a ) -1-1-onto-> ( Base `  b
) }  =  {
c  e.  ( R LMHom 
S )  |  c : B -1-1-onto-> C } )
141, 3, 13elovmpt2 6294 . 2  |-  ( F  e.  ( R LMIso  S
)  <->  ( R  e. 
LMod  /\  S  e.  LMod  /\  F  e.  { c  e.  ( R LMHom  S
)  |  c : B -1-1-onto-> C } ) )
15 df-3an 939 . 2  |-  ( ( R  e.  LMod  /\  S  e.  LMod  /\  F  e.  { c  e.  ( R LMHom 
S )  |  c : B -1-1-onto-> C } )  <->  ( ( R  e.  LMod  /\  S  e.  LMod )  /\  F  e.  { c  e.  ( R LMHom  S )  |  c : B -1-1-onto-> C }
) )
16 f1oeq1 5668 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
1716elrab 3094 . . . 4  |-  ( F  e.  { c  e.  ( R LMHom  S )  |  c : B -1-1-onto-> C } 
<->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C ) )
1817anbi2i 677 . . 3  |-  ( ( ( R  e.  LMod  /\  S  e.  LMod )  /\  F  e.  { c  e.  ( R LMHom  S
)  |  c : B -1-1-onto-> C } )  <->  ( ( R  e.  LMod  /\  S  e.  LMod )  /\  ( F  e.  ( R LMHom  S )  /\  F : B
-1-1-onto-> C ) ) )
19 lmhmlmod1 16114 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  R  e.  LMod )
20 lmhmlmod2 16113 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  S  e.  LMod )
2119, 20jca 520 . . . . 5  |-  ( F  e.  ( R LMHom  S
)  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2221adantr 453 . . . 4  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2322pm4.71ri 616 . . 3  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : B -1-1-onto-> C )  <->  ( ( R  e.  LMod  /\  S  e.  LMod )  /\  ( F  e.  ( R LMHom  S )  /\  F : B
-1-1-onto-> C ) ) )
2418, 23bitr4i 245 . 2  |-  ( ( ( R  e.  LMod  /\  S  e.  LMod )  /\  F  e.  { c  e.  ( R LMHom  S
)  |  c : B -1-1-onto-> C } )  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )
2514, 15, 243bitri 264 1  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   Basecbs 13474   LModclmod 15955   LMHom clmhm 16100   LMIso clmim 16101
This theorem is referenced by:  lmimf1o  16140  lmimlmhm  16141  islmim2  16143  pwssplit4  27181  indlcim  27300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-lmhm 16103  df-lmim 16104
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