Theorem f1o2 2804

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

Assertion

Ref	Expression
f1o2	|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

Proof of Theorem f1o2

Step	Hyp	Ref	Expression
1		df-f1 2435	. . . . . 6 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
2	1	pm3.27bd 263	. . . . 5 |- (F:A-1-1->B -> Fun `'F)
3		df-fo 2436	. . . . . 6 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	3	biimp 133	. . . . 5 |- (F:A-onto->B -> (F Fn A /\ ran F = B))
5	2, 4	anim12i 268	. . . 4 |- ((F:A-1-1->B /\ F:A-onto->B) -> (Fun `'F /\ (F Fn A /\ ran F = B)))
6		eqimss 1548	. . . . . . . . . 10 |- (ran F = B -> ran F (_ B)
7	6	anim2i 270	. . . . . . . . 9 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F (_ B))
8		df-f 2434	. . . . . . . . 9 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
9	7, 8	sylibr 175	. . . . . . . 8 |- ((F Fn A /\ ran F = B) -> F:A-->B)
10	9	anim1i 269	. . . . . . 7 |- (((F Fn A /\ ran F = B) /\ Fun `'F) -> (F:A-->B /\ Fun `'F))
11	10, 1	sylibr 175	. . . . . 6 |- (((F Fn A /\ ran F = B) /\ Fun `'F) -> F:A-1-1->B)
12	11	ancoms 334	. . . . 5 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> F:A-1-1->B)
13	3	biimpr 134	. . . . . 6 |- ((F Fn A /\ ran F = B) -> F:A-onto->B)
14	13	adantl 305	. . . . 5 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> F:A-onto->B)
15	12, 14	jca 236	. . . 4 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> (F:A-1-1->B /\ F:A-onto->B))
16	5, 15	impbi 139	. . 3 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (Fun `'F /\ (F Fn A /\ ran F = B)))
17		an12 370	. . 3 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
18	16, 17	bitr 151	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
19		df-f1o 2437	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
20		3anass 585	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
21	18, 19, 20	3bitr4 158	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

Syntax hints:   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091   (_ wss 1487  `'ccnv 2409  ran crn 2411  Fun wfun 2416   Fn wfn 2417  -->wf 2418  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421

This theorem is referenced by:  f1o4 2807  f1orn 2809  f1ocnv 2811  f1oco 2816  tz7.49c 2998  fiint 3445  infxpidmlem4 4936  infxpidmlem7 4939

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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