Theorem relrelss 5255

Description: Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Assertion

Ref	Expression
relrelss	|- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) )

Proof of Theorem relrelss

Step	Hyp	Ref	Expression
1		df-rel 4726	. . 3 |- ( Rel dom A <-> dom A C_ ( _V X. _V ) )
2	1	anbi2i 457	. 2 |- ( ( Rel A /\ Rel dom A ) <-> ( Rel A /\ dom A C_ ( _V X. _V ) ) )
3		relssdmrn 5249	. . . 4 |- ( Rel A -> A C_ ( dom A X. ran A ) )
4		ssv 3246	. . . . 5 |- ran A C_ _V
5		xpss12 4826	. . . . 5 |- ( ( dom A C_ ( _V X. _V ) /\ ran A C_ _V ) -> ( dom A X. ran A ) C_ ( ( _V X. _V ) X. _V ) )
6	4, 5	mpan2 425	. . . 4 |- ( dom A C_ ( _V X. _V ) -> ( dom A X. ran A ) C_ ( ( _V X. _V ) X. _V ) )
7	3, 6	sylan9ss 3237	. . 3 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) -> A C_ ( ( _V X. _V ) X. _V ) )
8		xpss 4827	. . . . . 6 |- ( ( _V X. _V ) X. _V ) C_ ( _V X. _V )
9		sstr 3232	. . . . . 6 |- ( ( A C_ ( ( _V X. _V ) X. _V ) /\ ( ( _V X. _V ) X. _V ) C_ ( _V X. _V ) ) -> A C_ ( _V X. _V ) )
10	8, 9	mpan2 425	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> A C_ ( _V X. _V ) )
11		df-rel 4726	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
12	10, 11	sylibr 134	. . . 4 |- ( A C_ ( ( _V X. _V ) X. _V ) -> Rel A )
13		dmss 4922	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ dom ( ( _V X. _V ) X. _V ) )
14		vn0m 3503	. . . . . 6 |- E. x x e. _V
15		dmxpm 4944	. . . . . 6 |- ( E. x x e. _V -> dom ( ( _V X. _V ) X. _V ) = ( _V X. _V ) )
16	14, 15	ax-mp 5	. . . . 5 |- dom ( ( _V X. _V ) X. _V ) = ( _V X. _V )
17	13, 16	sseqtrdi 3272	. . . 4 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ ( _V X. _V ) )
18	12, 17	jca 306	. . 3 |- ( A C_ ( ( _V X. _V ) X. _V ) -> ( Rel A /\ dom A C_ ( _V X. _V ) ) )
19	7, 18	impbii 126	. 2 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) <-> A C_ ( ( _V X. _V ) X. _V ) )
20	2, 19	bitri 184	1 |- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   C_ wss 3197   X. cxp 4717   dom cdm 4719   ran crn 4720   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  dftpos3  6408  tpostpos2  6411