Theorem svrelfun 5283

Description: A single-valued relation is a function. (See fun2cnv 5282 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
svrelfun	|- ( Fun A <-> ( Rel A /\ Fun `' `' A ) )

Proof of Theorem svrelfun

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun6 5232	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		fun2cnv 5282	. . 3 |- ( Fun `' `' A <-> A. x E* y x A y )
3	2	anbi2i 457	. 2 |- ( ( Rel A /\ Fun `' `' A ) <-> ( Rel A /\ A. x E* y x A y ) )
4	1, 3	bitr4i 187	1 |- ( Fun A <-> ( Rel A /\ Fun `' `' A ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1351   E*wmo 2027   class class class wbr 4005   `'ccnv 4627   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-rex 2461  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-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-fun 5220

This theorem is referenced by: (None)