Theorem dffun9 5196

Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
dffun9	|- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )

Distinct variable group:   x, y, A

Proof of Theorem dffun9

Step	Hyp	Ref	Expression
1		dffun7 5194	. 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2		vex 2715	. . . . . . . 8 |- x e. _V
3		vex 2715	. . . . . . . 8 |- y e. _V
4	2, 3	brelrn 4816	. . . . . . 7 |- ( x A y -> y e. ran A )
5	4	pm4.71ri 390	. . . . . 6 |- ( x A y <-> ( y e. ran A /\ x A y ) )
6	5	mobii 2043	. . . . 5 |- ( E* y x A y <-> E* y ( y e. ran A /\ x A y ) )
7		df-rmo 2443	. . . . 5 |- ( E* y e. ran A x A y <-> E* y ( y e. ran A /\ x A y ) )
8	6, 7	bitr4i 186	. . . 4 |- ( E* y x A y <-> E* y e. ran A x A y )
9	8	ralbii 2463	. . 3 |- ( A. x e. dom A E* y x A y <-> A. x e. dom A E* y e. ran A x A y )
10	9	anbi2i 453	. 2 |- ( ( Rel A /\ A. x e. dom A E* y x A y ) <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )
11	1, 10	bitri 183	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )

Syntax hints:   /\ wa 103   <-> wb 104   E*wmo 2007   e. wcel 2128   A.wral 2435   E*wrmo 2438   class class class wbr 3965   dom cdm 4583   ran crn 4584   Rel wrel 4588   Fun wfun 5161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rmo 2443  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4252  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-fun 5169

This theorem is referenced by: (None)