Theorem f1ocnv 5082

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 4940	. . . . 5 |- ( F Fn A -> Rel F)
2		dfrel2 4714	. . . . . 6 |- ( Rel F <-> `' `' F = F)
3		fneq1 4930	. . . . . . 7 |- ( `' `' F = F -> ( `' `' F Fn A <-> F Fn A))
4	3	biimprd 147	. . . . . 6 |- ( `' `' F = F -> ( F Fn A -> `' `' F Fn A))
5	2, 4	sylbi 114	. . . . 5 |- ( Rel F -> ( F Fn A -> `' `' F Fn A))
6	1, 5	mpcom 32	. . . 4 |- ( F Fn A -> `' `' F Fn A)
7	6	anim2i 324	. . 3 |- (( `' F Fn B /\ F Fn A) -> ( `' F Fn B /\ `' `' F Fn A))
8	7	ancoms 255	. 2 |- (( F Fn A /\ `' F Fn B) -> ( `' F Fn B /\ `' `' F Fn A))
9		dff1o4 5077	. 2 |- ( F :A -1-1-onto->B <-> ( F Fn A /\ `' F Fn B))
10		dff1o4 5077	. 2 |- ( `' F :B -1-1-onto->A <-> ( `' F Fn B /\ `' `' F Fn A))
11	8, 9, 10	3imtr4i 190	1 |- ( F :A -1-1-onto->B -> `' F :B -1-1-onto->A)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   `'ccnv 4287   Rel wrel 4293   Fn wfn 4840   -1-1-onto->wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  f1ocnvb  5083  f1orescnv  5085  f1imacnv  5086  f1cnv  5093  f1ococnv1  5098  f1oresrab  5272  f1ocnvfv2  5361  f1ocnvdm  5364  f1ocnvfvrneq  5365  fcof1o  5372  isocnv  5394  f1ofveu  5443  ener  6195  en0  6211  en1  6215  cnrecnv  9138