Theorem svrelfun 3560

Description: A single-valued relation is a function. (See fun2cnv 3559 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24.

Assertion

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

Proof of Theorem svrelfun

Step	Hyp	Ref	Expression
1		dffunmo 3531	. 2 |- (Fun A <-> (Rel A /\ A.xE*y xAy))
2		fun2cnv 3559	. . 3 |- (Fun `'`'A <-> A.xE*y xAy)
3	2	anbi2i 480	. 2 |- ((Rel A /\ Fun `'`'A) <-> (Rel A /\ A.xE*y xAy))
4	1, 3	bitr4 176	1 |- (Fun A <-> (Rel A /\ Fun `'`'A))

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 954  E*wmo 1381   class class class wbr 2619  `'ccnv 3169  Rel wrel 3175  Fun wfun 3176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-fun 3192

Copyright terms: Public domain