Theorem fdmrn 6037

Description: A different way to write F is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Assertion

Ref	Expression
fdmrn	|- ( Fun F <-> F : dom F --> ran F )

Proof of Theorem fdmrn

Step	Hyp	Ref	Expression
1		ssid 3268	. . 3 |- ran F C_ ran F
2		df-f 5384	. . 3 |- ( F : dom F --> ran F <-> ( F Fn dom F /\ ran F C_ ran F ) )
3	1, 2	mpbiran2 954	. 2 |- ( F : dom F --> ran F <-> F Fn dom F )
4		eqid 2238	. . 3 |- dom F = dom F
5		df-fn 5383	. . 3 |- ( F Fn dom F <-> ( Fun F /\ dom F = dom F ) )
6	4, 5	mpbiran2 954	. 2 |- ( F Fn dom F <-> Fun F )
7	3, 6	bitr2i 185	1 |- ( Fun F <-> F : dom F --> ran F )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   C_ wss 3220   dom cdm 4776   ran crn 4777   Fun wfun 5374   Fn wfn 5375   -->wf 5376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-fn 5383  df-f 5384

This theorem is used by:  rinvf1o  6038