Theorem f1ocnv 3686

Description: The converse of a one-to-one onto function is also one-to-one onto.

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		df-rn 3179	. . . . . . . 8 |- ran F = dom `' F
2	1	eqeq1i 1474	. . . . . . 7 |- (ran F = B <-> dom `' F = B)
3	2	anbi2i 479	. . . . . 6 |- ((Fun `'F /\ ran F = B) <-> (Fun `'F /\ dom `' F = B))
4		df-fn 3183	. . . . . 6 |- (`'F Fn B <-> (Fun `'F /\ dom `' F = B))
5	3, 4	bitr4 176	. . . . 5 |- ((Fun `'F /\ ran F = B) <-> `'F Fn B)
6	5	biimp 151	. . . 4 |- ((Fun `'F /\ ran F = B) -> `'F Fn B)
7		fnfun 3571	. . . . . 6 |- (F Fn A -> Fun F)
8		funcnvcnv 3541	. . . . . 6 |- (Fun F -> Fun `'`'F)
9	7, 8	syl 10	. . . . 5 |- (F Fn A -> Fun `'`'F)
10		fndm 3573	. . . . . 6 |- (F Fn A -> dom F = A)
11		dfdm4 3294	. . . . . 6 |- dom F = ran `' F
12	10, 11	syl5eqr 1513	. . . . 5 |- (F Fn A -> ran `' F = A)
13	9, 12	jca 288	. . . 4 |- (F Fn A -> (Fun `'`'F /\ ran `' F = A))
14	6, 13	anim12i 333	. . 3 |- (((Fun `'F /\ ran F = B) /\ F Fn A) -> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
15	14	ancoms 436	. 2 |- ((F Fn A /\ (Fun `'F /\ ran F = B)) -> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
16		f1o2 3678	. . 3 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
17		3anass 777	. . 3 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
18	16, 17	bitr 173	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
19		f1o2 3678	. . 3 |- (`'F:B-1-1-onto->A <-> (`'F Fn B /\ Fun `'`'F /\ ran `' F = A))
20		3anass 777	. . 3 |- ((`'F Fn B /\ Fun `'`'F /\ ran `' F = A) <-> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
21	19, 20	bitr 173	. 2 |- (`'F:B-1-1-onto->A <-> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
22	15, 18, 21	3imtr4 219	1 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953  `'ccnv 3159  dom cdm 3160  ran crn 3161  Fun wfun 3166   Fn wfn 3167  -1-1-onto->wf1o 3171

This theorem is referenced by:  f1ocnvb 3687  f1orescnv 3689  f1imacnv 3690  f1ococnv2 3693  f1ococnv1 3694  f1dmex 3695  f1ocnvfv1 3863  f1ocnvfv2 3864  f1ofveu 3867  f1ocnvfv3 3868  f1ocnvdm 3869  isocnv 3881  ener 4391  en0 4404  en1 4407  mapenlem2 4470  ssenen 4484  fodomfi 4540  weth 4759  uzrdgval 6239  uzrdgsuc 6241  uzrdgfnuz 6243  unbenlem 7447  effoi 8666  effoiOLD 8667  logrn 8673  logf1o 8677  dflog2OLD 8701  cnvunopt 9758  unopadjt 9759  symggrpiOLD 10311  symggrpi 10313  f2imacnv 10370  cnvhmpha 10412  hmphsyma 10415

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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