Theorem svrelfun 5446

Description: A single-valued relation is a function. (See fun2cnv 5445 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 5391	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		fun2cnv 5445	. . 3 |- ( Fun `' `' A <-> A. x E* y x A y )
3	2	anbi2i 461	. 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 ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   A.wal 1400   E*wmo 2087   class class class wbr 4130   `'ccnv 4773   Rel wrel 4779   Fun wfun 5371

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof 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-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-fun 5379

This theorem is used by: (None)