Theorem dff1o5OLD 4647

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

Assertion

Ref	Expression
dff1o5OLD	|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))

Proof of Theorem dff1o5OLD

Step	Hyp	Ref	Expression
1		df-f1o 4013	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		df-fo 4012	. . 3 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
3	2	anbi2i 538	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (F:A-1-1->B /\ (F Fn A /\ ran F = B)))
4		an12 542	. . 3 |- ((F:A-1-1->B /\ (F Fn A /\ ran F = B)) <-> (F Fn A /\ (F:A-1-1->B /\ ran F = B)))
5		f1f 4610	. . . . . 6 |- (F:A-1-1->B -> F:A-->B)
6		ffn 4562	. . . . . 6 |- (F:A-->B -> F Fn A)
7	5, 6	syl 12	. . . . 5 |- (F:A-1-1->B -> F Fn A)
8	7	adantr 425	. . . 4 |- ((F:A-1-1->B /\ ran F = B) -> F Fn A)
9	8	pm4.71ri 700	. . 3 |- ((F:A-1-1->B /\ ran F = B) <-> (F Fn A /\ (F:A-1-1->B /\ ran F = B)))
10	4, 9	bitr4i 193	. 2 |- ((F:A-1-1->B /\ (F Fn A /\ ran F = B)) <-> (F:A-1-1->B /\ ran F = B))
11	1, 3, 10	3bitri 194	1 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  ran crn 3987   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

This theorem depends on definitions:  df-bi 164  df-an 242  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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