Theorem dffun9 5227

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 5225	. 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2		vex 2733	. . . . . . . 8 |- x e. _V
3		vex 2733	. . . . . . . 8 |- y e. _V
4	2, 3	brelrn 4844	. . . . . . 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 2056	. . . . 5 |- ( E* y x A y <-> E* y ( y e. ran A /\ x A y ) )
7		df-rmo 2456	. . . . 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 2476	. . 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 454	. 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 2020   e. wcel 2141   A.wral 2448   E*wrmo 2451   class class class wbr 3989   dom cdm 4611   ran crn 4612   Rel wrel 4616   Fun wfun 5192

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rmo 2456  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200

This theorem is referenced by: (None)