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Theorem islmim 16025
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 15990 . . 3  |- LMIso  =  ( a  e.  LMod ,  b  e.  LMod  |->  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
2 ovex 6006 . . . 4  |-  ( a LMHom 
b )  e.  _V
32rabex 4267 . . 3  |-  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
4 oveq12 5990 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a LMHom  b )  =  ( R LMHom  S
) )
5 fveq2 5632 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
6 islmim.b . . . . . 6  |-  B  =  ( Base `  R
)
75, 6syl6eqr 2416 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
8 fveq2 5632 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
9 islmim.c . . . . . 6  |-  C  =  ( Base `  S
)
108, 9syl6eqr 2416 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
11 f1oeq23 5572 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
127, 10, 11syl2an 463 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
134, 12rabeqbidv 2868 . . 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 6191 . 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 937 . 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 5569 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
1716elrab 3009 . . . 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 675 . . 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 16000 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  R  e.  LMod )
20 lmhmlmod2 15999 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  S  e.  LMod )
2119, 20jca 518 . . . . 5  |-  ( F  e.  ( R LMHom  S
)  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2221adantr 451 . . . 4  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2322pm4.71ri 614 . . 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 243 . 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 262 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   {crab 2632   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981   Basecbs 13356   LModclmod 15837   LMHom clmhm 15986   LMIso clmim 15987
This theorem is referenced by:  lmimf1o  16026  lmimlmhm  16027  islmim2  16029  pwssplit4  26782  indlcim  26901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-lmhm 15989  df-lmim 15990
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