Theorem f1ocnv 5476

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 5316	. . . . 5 |- ( F Fn A -> Rel F )
2		dfrel2 5081	. . . . . 6 |- ( Rel F <-> `' `' F = F )
3		fneq1 5306	. . . . . . 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 5471	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
10		dff1o4 5471	. 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 1353   `'ccnv 4627   Rel wrel 4633   Fn wfn 5213   -1-1-onto->wf1o 5217

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225

This theorem is referenced by:  f1ocnvb  5477  f1orescnv  5479  f1imacnv  5480  f1cnv  5487  f1ococnv1  5492  f1oresrab  5683  f1ocnvfv2  5781  f1ocnvdm  5784  f1ocnvfvrneq  5785  fcof1o  5792  isocnv  5814  f1ofveu  5865  mapsnf1o3  6699  ener  6781  en0  6797  en1  6801  mapen  6848  ssenen  6853  preimaf1ofi  6952  ordiso2  7036  caseinl  7092  caseinr  7093  ctssdccl  7112  ctssdclemr  7113  enomnilem  7138  enmkvlem  7161  enwomnilem  7169  cc3  7269  fnn0nninf  10439  0tonninf  10441  1tonninf  10442  iseqf1olemkle  10486  iseqf1olemklt  10487  iseqf1olemqcl  10488  iseqf1olemnab  10490  iseqf1olemmo  10494  iseqf1olemqk  10496  seq3f1olemqsumkj  10500  seq3f1olemqsumk  10501  seq3f1olemstep  10503  hashfz1  10765  hashfacen  10818  seq3coll  10824  cnrecnv  10921  nnf1o  11386  summodclem3  11390  summodclem2a  11391  prodmodclem3  11585  prodmodclem2a  11586  fprodssdc  11600  sqpweven  12177  2sqpwodd  12178  phimullem  12227  eulerthlemh  12233  1arith2  12368  xpnnen  12397  ennnfonelemjn  12405  ennnfonelemp1  12409  ennnfonelemhdmp1  12412  ennnfonelemss  12413  ennnfonelemkh  12415  ennnfonelemhf1o  12416  ennnfonelemex  12417  ennnfonelemf1  12421  ennnfonelemnn0  12425  ennnfonelemim  12427  ctinfomlemom  12430  ctiunctlemfo  12442  ssnnctlemct  12449  mhmf1o  12866  txhmeo  13858  dfrelog  14320  relogf1o  14321  012of  14784  exmidsbthrlem  14809  iswomninnlem  14836