Theorem f1o5 3703

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

Assertion

Ref	Expression
f1o5	|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))

Proof of Theorem f1o5

Step	Hyp	Ref	Expression
1		df-f1o 3203	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		df-fo 3202	. . 3 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
3	2	anbi2i 482	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (F:A-1-1->B /\ (F Fn A /\ ran F = B)))
4		an12 486	. . 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 3671	. . . . . 6 |- (F:A-1-1->B -> F:A-->B)
6		ffn 3633	. . . . . 6 |- (F:A-->B -> F Fn A)
7	5, 6	syl 10	. . . . 5 |- (F:A-1-1->B -> F Fn A)
8	7	adantr 391	. . . 4 |- ((F:A-1-1->B /\ ran F = B) -> F Fn A)
9	8	pm4.71ri 640	. . 3 |- ((F:A-1-1->B /\ ran F = B) <-> (F Fn A /\ (F:A-1-1->B /\ ran F = B)))
10	4, 9	bitr4 176	. 2 |- ((F:A-1-1->B /\ (F Fn A /\ ran F = B)) <-> (F:A-1-1->B /\ ran F = B))
11	1, 3, 10	3bitr 177	1 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958  ran crn 3177   Fn wfn 3183  -->wf 3184  -1-1->wf1 3185  -onto->wfo 3186  -1-1-onto->wf1o 3187

This theorem is referenced by:  f1orescnv 3710  mapenlem2 4496  om2uzf1o 6302  grplactf1o 8094  logrn 8746  hmeogrp 10524  homcard 10525

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 147  df-an 225  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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