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Theorem f1ocnv 5475
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 5315 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel2 5080 . . . . . 6  |-  ( Rel 
F  <->  `' `' F  =  F
)
3 fneq1 5305 . . . . . . 7  |-  ( `' `' F  =  F  ->  ( `' `' F  Fn  A  <->  F  Fn  A
) )
43biimprd 158 . . . . . 6  |-  ( `' `' F  =  F  ->  ( F  Fn  A  ->  `' `' F  Fn  A
) )
52, 4sylbi 121 . . . . 5  |-  ( Rel 
F  ->  ( F  Fn  A  ->  `' `' F  Fn  A )
)
61, 5mpcom 36 . . . 4  |-  ( F  Fn  A  ->  `' `' F  Fn  A
)
76anim2i 342 . . 3  |-  ( ( `' F  Fn  B  /\  F  Fn  A
)  ->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
87ancoms 268 . 2  |-  ( ( F  Fn  A  /\  `' F  Fn  B
)  ->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
9 dff1o4 5470 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
10 dff1o4 5470 . 2  |-  ( `' F : B -1-1-onto-> A  <->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
118, 9, 103imtr4i 201 1  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   `'ccnv 4626   Rel wrel 4632    Fn wfn 5212   -1-1-onto->wf1o 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224
This theorem is referenced by:  f1ocnvb  5476  f1orescnv  5478  f1imacnv  5479  f1cnv  5486  f1ococnv1  5491  f1oresrab  5682  f1ocnvfv2  5779  f1ocnvdm  5782  f1ocnvfvrneq  5783  fcof1o  5790  isocnv  5812  f1ofveu  5863  mapsnf1o3  6697  ener  6779  en0  6795  en1  6799  mapen  6846  ssenen  6851  preimaf1ofi  6950  ordiso2  7034  caseinl  7090  caseinr  7091  ctssdccl  7110  ctssdclemr  7111  enomnilem  7136  enmkvlem  7159  enwomnilem  7167  cc3  7267  fnn0nninf  10437  0tonninf  10439  1tonninf  10440  iseqf1olemkle  10484  iseqf1olemklt  10485  iseqf1olemqcl  10486  iseqf1olemnab  10488  iseqf1olemmo  10492  iseqf1olemqk  10494  seq3f1olemqsumkj  10498  seq3f1olemqsumk  10499  seq3f1olemstep  10501  hashfz1  10763  hashfacen  10816  seq3coll  10822  cnrecnv  10919  nnf1o  11384  summodclem3  11388  summodclem2a  11389  prodmodclem3  11583  prodmodclem2a  11584  fprodssdc  11598  sqpweven  12175  2sqpwodd  12176  phimullem  12225  eulerthlemh  12231  1arith2  12366  xpnnen  12395  ennnfonelemjn  12403  ennnfonelemp1  12407  ennnfonelemhdmp1  12410  ennnfonelemss  12411  ennnfonelemkh  12413  ennnfonelemhf1o  12414  ennnfonelemex  12415  ennnfonelemf1  12419  ennnfonelemnn0  12423  ennnfonelemim  12425  ctinfomlemom  12428  ctiunctlemfo  12440  ssnnctlemct  12447  mhmf1o  12861  txhmeo  13822  dfrelog  14284  relogf1o  14285  012of  14748  exmidsbthrlem  14773  iswomninnlem  14800
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