Theorem f1ocnv 5159

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 5017	. . . . 5 |- ( F Fn A -> Rel F )
2		dfrel2 4791	. . . . . 6 |- ( Rel F <-> `' `' F = F )
3		fneq1 5007	. . . . . . 7 |- ( `' `' F = F -> ( `' `' F Fn A <-> F Fn A ) )
4	3	biimprd 156	. . . . . 6 |- ( `' `' F = F -> ( F Fn A -> `' `' F Fn A ) )
5	2, 4	sylbi 119	. . . . 5 |- ( Rel F -> ( F Fn A -> `' `' F Fn A ) )
6	1, 5	mpcom 36	. . . 4 |- ( F Fn A -> `' `' F Fn A )
7	6	anim2i 334	. . 3 |- ( ( `' F Fn B /\ F Fn A ) -> ( `' F Fn B /\ `' `' F Fn A ) )
8	7	ancoms 264	. 2 |- ( ( F Fn A /\ `' F Fn B ) -> ( `' F Fn B /\ `' `' F Fn A ) )
9		dff1o4 5154	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
10		dff1o4 5154	. 2 |- ( `' F : B -1-1-onto-> A <-> ( `' F Fn B /\ `' `' F Fn A ) )
11	8, 9, 10	3imtr4i 199	1 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   `'ccnv 4362   Rel wrel 4368   Fn wfn 4917   -1-1-onto->wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  f1ocnvb  5160  f1orescnv  5162  f1imacnv  5163  f1cnv  5170  f1ococnv1  5175  f1oresrab  5350  f1ocnvfv2  5438  f1ocnvdm  5441  f1ocnvfvrneq  5442  fcof1o  5449  isocnv  5471  f1ofveu  5520  ener  6282  en0  6298  en1  6302  ordiso2  6446  cnrecnv  9797  sqpweven  10553  2sqpwodd  10554