Theorem f1ocnv 5139

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 4997	. . . . 5 |- ( F Fn A -> Rel F )
2		dfrel2 4771	. . . . . 6 |- ( Rel F <-> `' `' F = F )
3		fneq1 4987	. . . . . . 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 5134	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
10		dff1o4 5134	. 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 1243   `'ccnv 4344   Rel wrel 4350   Fn wfn 4897   -1-1-onto->wf1o 4901

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1ocnvb  5140  f1orescnv  5142  f1imacnv  5143  f1cnv  5150  f1ococnv1  5155  f1oresrab  5329  f1ocnvfv2  5418  f1ocnvdm  5421  f1ocnvfvrneq  5422  fcof1o  5429  isocnv  5451  f1ofveu  5500  ener  6259  en0  6275  en1  6279  ordiso2  6357  cnrecnv  9510