Theorem dffun9 5401

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 5399	. 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2		vex 2824	. . . . . . . 8 |- x e. _V
3		vex 2824	. . . . . . . 8 |- y e. _V
4	2, 3	brelrn 5010	. . . . . . 7 |- ( x A y -> y e. ran A )
5	4	pm4.71ri 396	. . . . . 6 |- ( x A y <-> ( y e. ran A /\ x A y ) )
6	5	mobii 2123	. . . . 5 |- ( E* y x A y <-> E* y ( y e. ran A /\ x A y ) )
7		df-rmo 2536	. . . . 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 2556	. . 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 461	. 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 2087   e. wcel 2209   A.wral 2528   E*wrmo 2531   class class class wbr 4125   dom cdm 4769   ran crn 4770   Rel wrel 4774   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rmo 2536  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374

This theorem is referenced by: (None)