Theorem relrelss 4864

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 4370	. . 3 |- ( Rel dom A <-> dom A C_ ( _V X. _V ) )
2	1	anbi2i 444	. 2 |- ( ( Rel A /\ Rel dom A ) <-> ( Rel A /\ dom A C_ ( _V X. _V ) ) )
3		relssdmrn 4861	. . . 4 |- ( Rel A -> A C_ ( dom A X. ran A ) )
4		ssv 3019	. . . . 5 |- ran A C_ _V
5		xpss12 4463	. . . . 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 415	. . . 4 |- ( dom A C_ ( _V X. _V ) -> ( dom A X. ran A ) C_ ( ( _V X. _V ) X. _V ) )
7	3, 6	sylan9ss 3012	. . 3 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) -> A C_ ( ( _V X. _V ) X. _V ) )
8		xpss 4464	. . . . . 6 |- ( ( _V X. _V ) X. _V ) C_ ( _V X. _V )
9		sstr 3007	. . . . . 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 415	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> A C_ ( _V X. _V ) )
11		df-rel 4370	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
12	10, 11	sylibr 132	. . . 4 |- ( A C_ ( ( _V X. _V ) X. _V ) -> Rel A )
13		dmss 4552	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ dom ( ( _V X. _V ) X. _V ) )
14		vn0m 3259	. . . . . 6 |- E. x x e. _V
15		dmxpm 4573	. . . . . 6 |- ( E. x x e. _V -> dom ( ( _V X. _V ) X. _V ) = ( _V X. _V ) )
16	14, 15	ax-mp 7	. . . . 5 |- dom ( ( _V X. _V ) X. _V ) = ( _V X. _V )
17	13, 16	syl6sseq 3045	. . . 4 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ ( _V X. _V ) )
18	12, 17	jca 300	. . 3 |- ( A C_ ( ( _V X. _V ) X. _V ) -> ( Rel A /\ dom A C_ ( _V X. _V ) ) )
19	7, 18	impbii 124	. 2 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) <-> A C_ ( ( _V X. _V ) X. _V ) )
20	2, 19	bitri 182	1 |- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   C_ wss 2973   X. cxp 4361   dom cdm 4363   ran crn 4364   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  dftpos3  5900  tpostpos2  5903