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Theorem f1ocnv 5466
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 5306 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel2 5071 . . . . . 6  |-  ( Rel 
F  <->  `' `' F  =  F
)
3 fneq1 5296 . . . . . . 7  |-  ( `' `' F  =  F  ->  ( `' `' F  Fn  A  <->  F  Fn  A
) )
43biimprd 159 . . . . . 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 5461 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
10 dff1o4 5461 . 2  |-  ( `' F : B -1-1-onto-> A  <->  ( `' F  Fn  B  /\  `' `' F  Fn  A
) )
118, 9, 103imtr4i 202 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 4619   Rel wrel 4625    Fn wfn 5203   -1-1-onto->wf1o 5207
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215
This theorem is referenced by:  f1ocnvb  5467  f1orescnv  5469  f1imacnv  5470  f1cnv  5477  f1ococnv1  5482  f1oresrab  5673  f1ocnvfv2  5769  f1ocnvdm  5772  f1ocnvfvrneq  5773  fcof1o  5780  isocnv  5802  f1ofveu  5853  mapsnf1o3  6687  ener  6769  en0  6785  en1  6789  mapen  6836  ssenen  6841  preimaf1ofi  6940  ordiso2  7024  caseinl  7080  caseinr  7081  ctssdccl  7100  ctssdclemr  7101  enomnilem  7126  enmkvlem  7149  enwomnilem  7157  cc3  7242  fnn0nninf  10405  0tonninf  10407  1tonninf  10408  iseqf1olemkle  10452  iseqf1olemklt  10453  iseqf1olemqcl  10454  iseqf1olemnab  10456  iseqf1olemmo  10460  iseqf1olemqk  10462  seq3f1olemqsumkj  10466  seq3f1olemqsumk  10467  seq3f1olemstep  10469  hashfz1  10729  hashfacen  10783  seq3coll  10789  cnrecnv  10886  nnf1o  11351  summodclem3  11355  summodclem2a  11356  prodmodclem3  11550  prodmodclem2a  11551  fprodssdc  11565  sqpweven  12141  2sqpwodd  12142  phimullem  12191  eulerthlemh  12197  1arith2  12332  xpnnen  12361  ennnfonelemjn  12369  ennnfonelemp1  12373  ennnfonelemhdmp1  12376  ennnfonelemss  12377  ennnfonelemkh  12379  ennnfonelemhf1o  12380  ennnfonelemex  12381  ennnfonelemf1  12385  ennnfonelemnn0  12389  ennnfonelemim  12391  ctinfomlemom  12394  ctiunctlemfo  12406  ssnnctlemct  12413  mhmf1o  12723  txhmeo  13312  dfrelog  13774  relogf1o  13775  012of  14227  exmidsbthrlem  14253  iswomninnlem  14280
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