Theorem fdmrn 6064

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 3624	. . 3 |- ran F C_ ran F
2		df-f 5892	. . 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 2622	. . 3 |- dom F = dom F
5		df-fn 5891	. . 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 265	1 |- ( Fun F <-> F : dom F --> ran F )

Syntax hints:   <-> wb 196   = wceq 1483   C_ wss 3574   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-fn 5891  df-f 5892

This theorem is referenced by:  nvof1o  6536  umgrwwlks2on  26850  rinvf1o  29432  smatrcl  29862  locfinref  29908  fco3  39421  limccog  39852