Theorem relrelss 4844

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 4352	. . 3 |- ( Rel dom A <-> dom A C_ ( _V X. _V ) )
2	1	anbi2i 430	. 2 |- ( ( Rel A /\ Rel dom A ) <-> ( Rel A /\ dom A C_ ( _V X. _V ) ) )
3		relssdmrn 4841	. . . 4 |- ( Rel A -> A C_ ( dom A X. ran A ) )
4		ssv 2965	. . . . 5 |- ran A C_ _V
5		xpss12 4445	. . . . 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 401	. . . 4 |- ( dom A C_ ( _V X. _V ) -> ( dom A X. ran A ) C_ ( ( _V X. _V ) X. _V ) )
7	3, 6	sylan9ss 2958	. . 3 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) -> A C_ ( ( _V X. _V ) X. _V ) )
8		xpss 4446	. . . . . 6 |- ( ( _V X. _V ) X. _V ) C_ ( _V X. _V )
9		sstr 2953	. . . . . 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 401	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> A C_ ( _V X. _V ) )
11		df-rel 4352	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
12	10, 11	sylibr 137	. . . 4 |- ( A C_ ( ( _V X. _V ) X. _V ) -> Rel A )
13		dmss 4534	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ dom ( ( _V X. _V ) X. _V ) )
14		vn0m 3232	. . . . . 6 |- E. x x e. _V
15		dmxpm 4555	. . . . . 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 2991	. . . 4 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ ( _V X. _V ) )
18	12, 17	jca 290	. . 3 |- ( A C_ ( ( _V X. _V ) X. _V ) -> ( Rel A /\ dom A C_ ( _V X. _V ) ) )
19	7, 18	impbii 117	. 2 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) <-> A C_ ( ( _V X. _V ) X. _V ) )
20	2, 19	bitri 173	1 |- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   C_ wss 2917   X. cxp 4343   dom cdm 4345   ran crn 4346   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  dftpos3  5877  tpostpos2  5880