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Theorem f1cnv 5497
Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cnv  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)

Proof of Theorem f1cnv
StepHypRef Expression
1 f1f1orn 5483 . 2  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ocnv 5485 . 2  |-  ( F : A -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> A )
31, 2syl 15 1  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   `'ccnv 4688   ran crn 4690   -1-1->wf1 5252   -1-1-onto->wf1o 5254
This theorem is referenced by:  f1dmex  5751  fin1a2lem7  8032  diophrw  26838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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