Theorem relrelss 5192

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 4666	. . 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 5186	. . . 4 |- ( Rel A -> A C_ ( dom A X. ran A ) )
4		ssv 3201	. . . . 5 |- ran A C_ _V
5		xpss12 4766	. . . . 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 3192	. . 3 |- ( ( Rel A /\ dom A C_ ( _V X. _V ) ) -> A C_ ( ( _V X. _V ) X. _V ) )
8		xpss 4767	. . . . . 6 |- ( ( _V X. _V ) X. _V ) C_ ( _V X. _V )
9		sstr 3187	. . . . . 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 4666	. . . . 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 4861	. . . . 5 |- ( A C_ ( ( _V X. _V ) X. _V ) -> dom A C_ dom ( ( _V X. _V ) X. _V ) )
14		vn0m 3458	. . . . . 6 |- E. x x e. _V
15		dmxpm 4882	. . . . . 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 3227	. . . 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 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   C_ wss 3153   X. cxp 4657   dom cdm 4659   ran crn 4660   Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  dftpos3  6315  tpostpos2  6318