Theorem f1ocnvOLD 4652

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

Assertion

Ref	Expression
f1ocnvOLD	|- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)

Proof of Theorem f1ocnvOLD

Step	Hyp	Ref	Expression
1		df-rn 4005	. . . . . . . 8 |- ran F = dom `' F
2	1	eqeq1i 1891	. . . . . . 7 |- (ran F = B <-> dom `' F = B)
3	2	anbi2i 538	. . . . . 6 |- ((Fun `'F /\ ran F = B) <-> (Fun `'F /\ dom `' F = B))
4		df-fn 4009	. . . . . 6 |- (`'F Fn B <-> (Fun `'F /\ dom `' F = B))
5	3, 4	bitr4i 193	. . . . 5 |- ((Fun `'F /\ ran F = B) <-> `'F Fn B)
6	5	biimpi 168	. . . 4 |- ((Fun `'F /\ ran F = B) -> `'F Fn B)
7		fnfun 4510	. . . . . 6 |- (F Fn A -> Fun F)
8		funcnvcnv 4473	. . . . . 6 |- (Fun F -> Fun `'`'F)
9	7, 8	syl 12	. . . . 5 |- (F Fn A -> Fun `'`'F)
10		fndm 4512	. . . . . 6 |- (F Fn A -> dom F = A)
11		dfdm4 4151	. . . . . 6 |- dom F = ran `' F
12	10, 11	syl5eqr 1942	. . . . 5 |- (F Fn A -> ran `' F = A)
13	9, 12	jca 310	. . . 4 |- (F Fn A -> (Fun `'`'F /\ ran `' F = A))
14	6, 13	anim12i 360	. . 3 |- (((Fun `'F /\ ran F = B) /\ F Fn A) -> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
15	14	ancoms 484	. 2 |- ((F Fn A /\ (Fun `'F /\ ran F = B)) -> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
16		dff1o2 4639	. . 3 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
17		3anass 862	. . 3 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
18	16, 17	bitri 190	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
19		dff1o2 4639	. . 3 |- (`'F:B-1-1-onto->A <-> (`'F Fn B /\ Fun `'`'F /\ ran `' F = A))
20		3anass 862	. . 3 |- ((`'F Fn B /\ Fun `'`'F /\ ran `' F = A) <-> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
21	19, 20	bitri 190	. 2 |- (`'F:B-1-1-onto->A <-> (`'F Fn B /\ (Fun `'`'F /\ ran `' F = A)))
22	15, 18, 21	3imtr4i 236	1 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298  `'ccnv 3985  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -1-1-onto->wf1o 3997

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

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