Theorem relrelss 5228

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 4700	. . 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 5222	. . . 4 |- ( Rel A -> A C_ ( dom A X. ran A ) )
4		ssv 3223	. . . . 5 |- ran A C_ _V
5		xpss12 4800	. . . . 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 3214	. . 3 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) -> A C_ ( ( _V X. _V ) X. _V ) )
8		xpss 4801	. . . . . 6 |- ( ( _V X. _V ) X. _V ) C_ ( _V X. _V )
9		sstr 3209	. . . . . 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 4700	. . . . 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 4896	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ dom ( ( _V X. _V ) X. _V ) )
14		vn0m 3480	. . . . . 6 |- E. x x e. _V
15		dmxpm 4917	. . . . . 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 3249	. . . 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 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   C_ wss 3174   X. cxp 4691   dom cdm 4693   ran crn 4694   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  dftpos3  6371  tpostpos2  6374