Theorem f1ocnv 5486

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

Step	Hyp	Ref	Expression
1		fnrel 5326	. . . . 5 |- ( F Fn A -> Rel F )
2		dfrel2 5091	. . . . . 6 |- ( Rel F <-> `' `' F = F )
3		fneq1 5316	. . . . . . 7 |- ( `' `' F = F -> ( `' `' F Fn A <-> F Fn A ) )
4	3	biimprd 158	. . . . . 6 |- ( `' `' F = F -> ( F Fn A -> `' `' F Fn A ) )
5	2, 4	sylbi 121	. . . . 5 |- ( Rel F -> ( F Fn A -> `' `' F Fn A ) )
6	1, 5	mpcom 36	. . . 4 |- ( F Fn A -> `' `' F Fn A )
7	6	anim2i 342	. . 3 |- ( ( `' F Fn B /\ F Fn A ) -> ( `' F Fn B /\ `' `' F Fn A ) )
8	7	ancoms 268	. 2 |- ( ( F Fn A /\ `' F Fn B ) -> ( `' F Fn B /\ `' `' F Fn A ) )
9		dff1o4 5481	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
10		dff1o4 5481	. 2 |- ( `' F : B -1-1-onto-> A <-> ( `' F Fn B /\ `' `' F Fn A ) )
11	8, 9, 10	3imtr4i 201	1 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   `'ccnv 4637   Rel wrel 4643   Fn wfn 5223   -1-1-onto->wf1o 5227

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235

This theorem is referenced by:  f1ocnvb  5487  f1orescnv  5489  f1imacnv  5490  f1cnv  5497  f1ococnv1  5502  f1oresrab  5694  f1ocnvfv2  5792  f1ocnvdm  5795  f1ocnvfvrneq  5796  fcof1o  5803  isocnv  5825  f1ofveu  5876  mapsnf1o3  6710  ener  6792  en0  6808  en1  6812  mapen  6859  ssenen  6864  preimaf1ofi  6963  ordiso2  7047  caseinl  7103  caseinr  7104  ctssdccl  7123  ctssdclemr  7124  enomnilem  7149  enmkvlem  7172  enwomnilem  7180  cc3  7280  fnn0nninf  10450  0tonninf  10452  1tonninf  10453  iseqf1olemkle  10497  iseqf1olemklt  10498  iseqf1olemqcl  10499  iseqf1olemnab  10501  iseqf1olemmo  10505  iseqf1olemqk  10507  seq3f1olemqsumkj  10511  seq3f1olemqsumk  10512  seq3f1olemstep  10514  hashfz1  10776  hashfacen  10829  seq3coll  10835  cnrecnv  10932  nnf1o  11397  summodclem3  11401  summodclem2a  11402  prodmodclem3  11596  prodmodclem2a  11597  fprodssdc  11611  sqpweven  12188  2sqpwodd  12189  phimullem  12238  eulerthlemh  12244  1arith2  12379  xpnnen  12408  ennnfonelemjn  12416  ennnfonelemp1  12420  ennnfonelemhdmp1  12423  ennnfonelemss  12424  ennnfonelemkh  12426  ennnfonelemhf1o  12427  ennnfonelemex  12428  ennnfonelemf1  12432  ennnfonelemnn0  12436  ennnfonelemim  12438  ctinfomlemom  12441  ctiunctlemfo  12453  ssnnctlemct  12460  mhmf1o  12882  txhmeo  14090  dfrelog  14552  relogf1o  14553  012of  15017  exmidsbthrlem  15042  iswomninnlem  15069