Theorem f1f1orn 3699

Description: A one-to-one function maps one-to-one onto its range.

Assertion

Ref	Expression
f1f1orn	|- (F:A-1-1->B -> F:A-1-1-onto->ran F)

Proof of Theorem f1f1orn

Step	Hyp	Ref	Expression
1		f1f 3665	. . . 4 |- (F:A-1-1->B -> F:A-->B)
2		ffn 3627	. . . 4 |- (F:A-->B -> F Fn A)
3	1, 2	syl 10	. . 3 |- (F:A-1-1->B -> F Fn A)
4		df-f1 3195	. . . 4 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
5	4	pm3.27bi 326	. . 3 |- (F:A-1-1->B -> Fun `'F)
6	3, 5	jca 288	. 2 |- (F:A-1-1->B -> (F Fn A /\ Fun `'F))
7		f1orn 3698	. 2 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))
8	6, 7	sylibr 200	1 |- (F:A-1-1->B -> F:A-1-1-onto->ran F)

Syntax hints:   -> wi 3   /\ wa 223  `'ccnv 3169  ran crn 3171  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -1-1->wf1 3179  -1-1-onto->wf1o 3181

This theorem is referenced by:  f1dmex 3710

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-3an 777  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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