Theorem dffun9 5259

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 5257	. 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2		vex 2754	. . . . . . . 8 |- x e. _V
3		vex 2754	. . . . . . . 8 |- y e. _V
4	2, 3	brelrn 4874	. . . . . . 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 2074	. . . . 5 |- ( E* y x A y <-> E* y ( y e. ran A /\ x A y ) )
7		df-rmo 2475	. . . . 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 2495	. . 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 2038   e. wcel 2159   A.wral 2467   E*wrmo 2470   class class class wbr 4017   dom cdm 4640   ran crn 4641   Rel wrel 4645   Fun wfun 5224

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rmo 2475  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-id 4307  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-fun 5232

This theorem is referenced by: (None)