Theorem f1ocnv 4651

Description: The converse of a one-to-one onto function is also one-to-one onto. (The proof was 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 4511	. . . . 5 |- (F Fn A -> Rel F)
2		dfrel2 4358	. . . . . 6 |- (Rel F <-> `'`'F = F)
3		fneq1 4503	. . . . . . 7 |- (`'`'F = F -> (`'`'F Fn A <-> F Fn A))
4	3	biimprd 171	. . . . . 6 |- (`'`'F = F -> (F Fn A -> `'`'F Fn A))
5	2, 4	sylbi 216	. . . . 5 |- (Rel F -> (F Fn A -> `'`'F Fn A))
6	1, 5	mpcom 60	. . . 4 |- (F Fn A -> `'`'F Fn A)
7	6	anim2i 362	. . 3 |- ((`'F Fn B /\ F Fn A) -> (`'F Fn B /\ `'`'F Fn A))
8	7	ancoms 484	. 2 |- ((F Fn A /\ `'F Fn B) -> (`'F Fn B /\ `'`'F Fn A))
9		dff1o4 4644	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
10		dff1o4 4644	. 2 |- (`'F:B-1-1-onto->A <-> (`'F Fn B /\ `'`'F Fn A))
11	8, 9, 10	3imtr4i 236	1 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298  `'ccnv 3985  Rel wrel 3991   Fn wfn 3993  -1-1-onto->wf1o 3997

This theorem is referenced by:  f1ocnvb 4653  f1orescnv 4655  f1imacnv 4656  f1ococnv2 4662  f1ococnv1 4663  f1dmex 4664  f1ocnvfv1 4854  f1ocnvfv2 4855  f1ofveu 4858  f1ocnvfv3 4859  f1ocnvdm 4860  isocnv 4873  ener 5469  en0 5482  en1 5485  ac6sfi 5509  mapenlem2 5584  ssenen 5598  fodomfi 5656  weth 5949  uzrdgvali 7714  uzrdgsuci 7716  uzrdgfnuzi 7718  unbenlem 8773  effoi 10099  logrn 10105  logf1o 10109  symggrpi 10205  hmeobc 10239  cnvhmpha 10240  fbssint 10279  comptoppr 10332  cnvunop 11479  unopadj 11480  f2imacnv 14355  f1ofi 14376  cnvinj 14463  domrancur1c 14550  hmphsyma 14882  finsschain 15373  opncldf3 15404  compfipin0lem 15435  compfipin0 15436  conntoppr 15445  fcluscomplem 15620  f1ocan1fv 15717  f1ocan2fv 15718  hmeocnv 15898  ismtycnv 15949  ismtyhmeo 15951  rngisocnv 16135

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

Copyright terms: Public domain