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Theorem lmimf1o 16137
Description: An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
islmim.b  |-  B  =  ( Base `  R
)
islmim.c  |-  C  =  ( Base `  S
)
Assertion
Ref Expression
lmimf1o  |-  ( F  e.  ( R LMIso  S
)  ->  F : B
-1-1-onto-> C )

Proof of Theorem lmimf1o
StepHypRef Expression
1 islmim.b . . 3  |-  B  =  ( Base `  R
)
2 islmim.c . . 3  |-  C  =  ( Base `  S
)
31, 2islmim 16136 . 2  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )
43simprbi 452 1  |-  ( F  e.  ( R LMIso  S
)  ->  F : B
-1-1-onto-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Basecbs 13471   LMHom clmhm 16097   LMIso clmim 16098
This theorem is referenced by:  lmimgim  16139  lnmlmic  27165  lmimlbs  27285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
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 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-lmhm 16100  df-lmim 16101
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