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Theorem f1ocnv 5380
Description: The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
f1ocnv  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )

Proof of Theorem f1ocnv
StepHypRef Expression
1 fnrel 5221 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel2 4989 . . . . . 6  |-  ( Rel 
F  <->  `' `' F  =  F
)
3 fneq1 5211 . . . . . . 7  |-  ( `' `' F  =  F  ->  ( `' `' F  Fn  A  <->  F  Fn  A
) )
43biimprd 157 . . . . . 6  |-  ( `' `' F  =  F  ->  ( F  Fn  A  ->  `' `' F  Fn  A
) )
52, 4sylbi 120 . . . . 5  |-  ( Rel 
F  ->  ( F  Fn  A  ->  `' `' F  Fn  A )
)
61, 5mpcom 36 . . . 4  |-  ( F  Fn  A  ->  `' `' F  Fn  A
)
76anim2i 339 . . 3  |-  ( ( `' F  Fn  B  /\  F  Fn  A
)  ->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
87ancoms 266 . 2  |-  ( ( F  Fn  A  /\  `' F  Fn  B
)  ->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
9 dff1o4 5375 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
10 dff1o4 5375 . 2  |-  ( `' F : B -1-1-onto-> A  <->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
118, 9, 103imtr4i 200 1  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   `'ccnv 4538   Rel wrel 4544    Fn wfn 5118   -1-1-onto->wf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  f1ocnvb  5381  f1orescnv  5383  f1imacnv  5384  f1cnv  5391  f1ococnv1  5396  f1oresrab  5585  f1ocnvfv2  5679  f1ocnvdm  5682  f1ocnvfvrneq  5683  fcof1o  5690  isocnv  5712  f1ofveu  5762  mapsnf1o3  6591  ener  6673  en0  6689  en1  6693  mapen  6740  ssenen  6745  preimaf1ofi  6839  ordiso2  6920  caseinl  6976  caseinr  6977  ctssdccl  6996  ctssdclemr  6997  enomnilem  7010  cc3  7083  fnn0nninf  10217  0tonninf  10219  1tonninf  10220  iseqf1olemkle  10264  iseqf1olemklt  10265  iseqf1olemqcl  10266  iseqf1olemnab  10268  iseqf1olemmo  10272  iseqf1olemqk  10274  seq3f1olemqsumkj  10278  seq3f1olemqsumk  10279  seq3f1olemstep  10281  hashfz1  10536  hashfacen  10586  seq3coll  10592  cnrecnv  10689  nnf1o  11152  summodclem3  11156  summodclem2a  11157  prodmodclem3  11351  prodmodclem2a  11352  sqpweven  11860  2sqpwodd  11861  phimullem  11908  xpnnen  11914  ennnfonelemjn  11922  ennnfonelemp1  11926  ennnfonelemhdmp1  11929  ennnfonelemss  11930  ennnfonelemkh  11932  ennnfonelemhf1o  11933  ennnfonelemex  11934  ennnfonelemf1  11938  ennnfonelemnn0  11942  ennnfonelemim  11944  ctinfomlemom  11947  ctiunctlemfo  11959  txhmeo  12498  dfrelog  12952  relogf1o  12953  exmidsbthrlem  13247  isomninnlem  13255
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