Theorem f1ocnv 3692

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 3184	. . . . . . . 8 |- ran F = dom `' F
2	1	eqeq1i 1479	. . . . . . 7 |- (ran F = B <-> dom `' F = B)
3	2	anbi2i 480	. . . . . 6 |- ((Fun `'F /\ ran F = B) <-> (Fun `'F /\ dom `' F = B))
4		df-fn 3188	. . . . . 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 3577	. . . . . 6 |- (F Fn A -> Fun F)
8		funcnvcnv 3547	. . . . . 6 |- (Fun F -> Fun `'`'F)
9	7, 8	syl 10	. . . . 5 |- (F Fn A -> Fun `'`'F)
10		fndm 3579	. . . . . 6 |- (F Fn A -> dom F = A)
11		dfdm4 3300	. . . . . 6 |- dom F = ran `' F
12	10, 11	syl5eqr 1518	. . . . 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 3684	. . 3 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
17		3anass 778	. . 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 3684	. . 3 |- (`'F:B-1-1-onto->A <-> (`'F Fn B /\ Fun `'`'F /\ ran `' F = A))
20		3anass 778	. . 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 774   = wceq 954  `'ccnv 3164  dom cdm 3165  ran crn 3166  Fun wfun 3171   Fn wfn 3172  -1-1-onto->wf1o 3176

This theorem is referenced by:  f1ocnvb 3693  f1orescnv 3695  f1imacnv 3696  f1ococnv2 3699  f1ococnv1 3700  f1dmex 3701  f1ocnvfv1 3869  f1ocnvfv2 3870  f1ofveu 3873  f1ocnvfv3 3874  f1ocnvdm 3875  isocnv 3887  ener 4397  en0 4410  en1 4413  mapenlem2 4476  ssenen 4490  fodomfi 4546  weth 4767  uzrdgval 6247  uzrdgsuc 6249  uzrdgfnuz 6251  unbenlem 7455  effoi 8684  logrn 8690  logf1o 8694  cnvunopt 9781  unopadjt 9782  symggrpi 10340  f2imacnv 10406  cnvhmpha 10448  hmphsyma 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192

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