Theorem relssdmrn 5124

Description: A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
relssdmrn	|- ( Rel A -> A C_ ( dom A X. ran A ) )

Proof of Theorem relssdmrn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( Rel A -> Rel A )
2		19.8a 1578	. . . 4 |- ( <. x , y >. e. A -> E. y <. x , y >. e. A )
3		19.8a 1578	. . . 4 |- ( <. x , y >. e. A -> E. x <. x , y >. e. A )
4		opelxp 4634	. . . . 5 |- ( <. x , y >. e. ( dom A X. ran A ) <-> ( x e. dom A /\ y e. ran A ) )
5		vex 2729	. . . . . . 7 |- x e. _V
6	5	eldm2 4802	. . . . . 6 |- ( x e. dom A <-> E. y <. x , y >. e. A )
7		vex 2729	. . . . . . 7 |- y e. _V
8	7	elrn2 4846	. . . . . 6 |- ( y e. ran A <-> E. x <. x , y >. e. A )
9	6, 8	anbi12i 456	. . . . 5 |- ( ( x e. dom A /\ y e. ran A ) <-> ( E. y <. x , y >. e. A /\ E. x <. x , y >. e. A ) )
10	4, 9	bitri 183	. . . 4 |- ( <. x , y >. e. ( dom A X. ran A ) <-> ( E. y <. x , y >. e. A /\ E. x <. x , y >. e. A ) )
11	2, 3, 10	sylanbrc 414	. . 3 |- ( <. x , y >. e. A -> <. x , y >. e. ( dom A X. ran A ) )
12	11	a1i 9	. 2 |- ( Rel A -> ( <. x , y >. e. A -> <. x , y >. e. ( dom A X. ran A ) ) )
13	1, 12	relssdv 4696	1 |- ( Rel A -> A C_ ( dom A X. ran A ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   C_ wss 3116   <.cop 3579   X. cxp 4602   dom cdm 4604   ran crn 4605   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  cnvssrndm  5125  cossxp  5126  relrelss  5130  relfld  5132  cnvexg  5141  fssxp  5355  oprabss  5928  resfunexgALT  6076  cofunexg  6077  fnexALT  6079  funexw  6080  erssxp  6524