Theorem f1o3 3694

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 485	. . 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 3195	. . . 4 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
3		df-fo 3196	. . . 4 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
4	2, 3	anbi12i 482	. . 3 |- ((F:A-1-1->B /\ F:A-onto->B) <-> ((F:A-->B /\ Fun `'F) /\ (F Fn A /\ ran F = B)))
5		eqimss 2109	. . . . . . 7 |- (ran F = B -> ran F (_ B)
6	5	anim2i 335	. . . . . 6 |- ((F Fn A /\ ran F = B) -> (F Fn A /\ ran F (_ B))
7		df-f 3194	. . . . . 6 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
8	6, 7	sylibr 200	. . . . 5 |- ((F Fn A /\ ran F = B) -> F:A-->B)
9	8	pm4.71ri 638	. . . 4 |- ((F Fn A /\ ran F = B) <-> (F:A-->B /\ (F Fn A /\ ran F = B)))
10	9	anbi1i 481	. . 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 183	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
12		df-f1o 3197	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
13	3	anbi1i 481	. 2 |- ((F:A-onto->B /\ Fun `'F) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
14	11, 12, 13	3bitr4 183	1 |- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   (_ wss 2047  `'ccnv 3169  ran crn 3171  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181

This theorem is referenced by:  f1ofo 3695  f1ores 3703  f11o 3712  f1oi 3717  2ndconst 4097  curry1 4098  ssdomg 4408  mapenlem1 4489  phplem4 4511  php3 4515  php3OLD 4516  relogf1o 8757

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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