Theorem dff1o2OLD 4640

Description: Alternate definition of one-to-one onto function.

Assertion

Ref	Expression
dff1o2OLD	|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

Proof of Theorem dff1o2OLD

Step	Hyp	Ref	Expression
1		df-f1 4011	. . . . . 6 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
2	1	simprbi 353	. . . . 5 |- (F:A-1-1->B -> Fun `'F)
3		df-fo 4012	. . . . . 6 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	3	biimpi 168	. . . . 5 |- (F:A-onto->B -> (F Fn A /\ ran F = B))
5	2, 4	anim12i 360	. . . 4 |- ((F:A-1-1->B /\ F:A-onto->B) -> (Fun `'F /\ (F Fn A /\ ran F = B)))
6		eqimss 2665	. . . . . . . . . 10 |- (ran F = B -> ran F C_ B)
7	6	anim2i 362	. . . . . . . . 9 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F C_ B))
8		df-f 4010	. . . . . . . . 9 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
9	7, 8	sylibr 217	. . . . . . . 8 |- ((F Fn A /\ ran F = B) -> F:A-->B)
10	9	anim1i 361	. . . . . . 7 |- (((F Fn A /\ ran F = B) /\ Fun `'F) -> (F:A-->B /\ Fun `'F))
11	10, 1	sylibr 217	. . . . . 6 |- (((F Fn A /\ ran F = B) /\ Fun `'F) -> F:A-1-1->B)
12	11	ancoms 484	. . . . 5 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> F:A-1-1->B)
13	3	biimpri 169	. . . . . 6 |- ((F Fn A /\ ran F = B) -> F:A-onto->B)
14	13	adantl 424	. . . . 5 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> F:A-onto->B)
15	12, 14	jca 310	. . . 4 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> (F:A-1-1->B /\ F:A-onto->B))
16	5, 15	impbii 174	. . 3 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (Fun `'F /\ (F Fn A /\ ran F = B)))
17		an12 542	. . 3 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
18	16, 17	bitri 190	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
19		df-f1o 4013	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
20		3anass 862	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
21	18, 19, 20	3bitr4i 200	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   C_ wss 2593  `'ccnv 3985  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996  -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-10 1308  ax-12 1310  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

This theorem depends on definitions:  df-bi 164  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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