Theorem svrelfun 5311

Description: A single-valued relation is a function. (See fun2cnv 5310 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 5260	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		fun2cnv 5310	. . 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 1362   E*wmo 2043   class class class wbr 4029   `'ccnv 4654   Rel wrel 4660   Fun wfun 5240

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-fun 5248

This theorem is referenced by: (None)