Theorem dffun9 5247

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 5245	. 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2		vex 2742	. . . . . . . 8 |- x e. _V
3		vex 2742	. . . . . . . 8 |- y e. _V
4	2, 3	brelrn 4862	. . . . . . 7 |- ( x A y -> y e. ran A )
5	4	pm4.71ri 392	. . . . . 6 |- ( x A y <-> ( y e. ran A /\ x A y ) )
6	5	mobii 2063	. . . . 5 |- ( E* y x A y <-> E* y ( y e. ran A /\ x A y ) )
7		df-rmo 2463	. . . . 5 |- ( E* y e. ran A x A y <-> E* y ( y e. ran A /\ x A y ) )
8	6, 7	bitr4i 187	. . . 4 |- ( E* y x A y <-> E* y e. ran A x A y )
9	8	ralbii 2483	. . 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 457	. 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 184	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )

Syntax hints:   /\ wa 104   <-> wb 105   E*wmo 2027   e. wcel 2148   A.wral 2455   E*wrmo 2458   class class class wbr 4005   dom cdm 4628   ran crn 4629   Rel wrel 4633   Fun wfun 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rmo 2463  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220

This theorem is referenced by: (None)