Theorem f1o2 3678

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 3185	. . . . . 6 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
2	1	pm3.27bi 326	. . . . 5 |- (F:A-1-1->B -> Fun `'F)
3		df-fo 3186	. . . . . 6 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	3	biimp 151	. . . . 5 |- (F:A-onto->B -> (F Fn A /\ ran F = B))
5	2, 4	anim12i 333	. . . 4 |- ((F:A-1-1->B /\ F:A-onto->B) -> (Fun `'F /\ (F Fn A /\ ran F = B)))
6		eqimss 2099	. . . . . . . . . 10 |- (ran F = B -> ran F (_ B)
7	6	anim2i 335	. . . . . . . . 9 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F (_ B))
8		df-f 3184	. . . . . . . . 9 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
9	7, 8	sylibr 200	. . . . . . . 8 |- ((F Fn A /\ ran F = B) -> F:A-->B)
10	9	anim1i 334	. . . . . . 7 |- (((F Fn A /\ ran F = B) /\ Fun `'F) -> (F:A-->B /\ Fun `'F))
11	10, 1	sylibr 200	. . . . . 6 |- (((F Fn A /\ ran F = B) /\ Fun `'F) -> F:A-1-1->B)
12	11	ancoms 436	. . . . 5 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> F:A-1-1->B)
13	3	biimpr 152	. . . . . 6 |- ((F Fn A /\ ran F = B) -> F:A-onto->B)
14	13	adantl 388	. . . . 5 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> F:A-onto->B)
15	12, 14	jca 288	. . . 4 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) -> (F:A-1-1->B /\ F:A-onto->B))
16	5, 15	impbi 157	. . 3 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (Fun `'F /\ (F Fn A /\ ran F = B)))
17		an12 483	. . 3 |- ((Fun `'F /\ (F Fn A /\ ran F = B)) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
18	16, 17	bitr 173	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
19		df-f1o 3187	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
20		3anass 777	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
21	18, 19, 20	3bitr4 183	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   (_ wss 2037  `'ccnv 3159  ran crn 3161  Fun wfun 3166   Fn wfn 3167  -->wf 3168  -1-1->wf1 3169  -onto->wfo 3170  -1-1-onto->wf1o 3171

This theorem is referenced by:  f1o4 3681  f1orn 3683  f1ocnv 3686  tz7.49c 3945  fiint 4534  infxpidmlem4 7498  infxpidmlem7 7501  dfrelog 8678  adj1o 9735  bra11 9954

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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