Theorem f1o3 2805

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

Assertion

Ref	Expression
f1o3	|- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))

Proof of Theorem f1o3

Step	Hyp	Ref	Expression
1		an23 371	. . 3 |- (((F:A-->B /\ Fun `'F) /\ (F Fn A /\ ran F = B)) <-> ((F:A-->B /\ (F Fn A /\ ran F = B)) /\ Fun `'F))
2		df-f1 2435	. . . 4 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
3		df-fo 2436	. . . 4 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	2, 3	anbi12i 369	. . 3 |- ((F:A-1-1->B /\ F:A-onto->B) <-> ((F:A-->B /\ Fun `'F) /\ (F Fn A /\ ran F = B)))
5		eqimss 1548	. . . . . . 7 |- (ran F = B -> ran F (_ B)
6	5	anim2i 270	. . . . . 6 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F (_ B))
7		df-f 2434	. . . . . 6 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
8	6, 7	sylibr 175	. . . . 5 |- ((F Fn A /\ ran F = B) -> F:A-->B)
9	8	pm4.71ri 484	. . . 4 |- ((F Fn A /\ ran F = B) <-> (F:A-->B /\ (F Fn A /\ ran F = B)))
10	9	anbi1i 368	. . 3 |- (((F Fn A /\ ran F = B) /\ Fun `'F) <-> ((F:A-->B /\ (F Fn A /\ ran F = B)) /\ Fun `'F))
11	1, 4, 10	3bitr4 158	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
12		df-f1o 2437	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
13	3	anbi1i 368	. 2 |- ((F:A-onto->B /\ Fun `'F) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
14	11, 12, 13	3bitr4 158	1 |- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))

Syntax hints:   <-> wb 127   /\ wa 196   = 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:  f1ofo 2806  f1ores 2813  f11o 2821  f1oi 2825  ssdomg 3311  mapenlem1 3384  phplem5 3407  php3 3411

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-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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