Theorem relssdmrn 5288

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 1639	. . . 4 |- ( <. x , y >. e. A -> E. y <. x , y >. e. A )
3		19.8a 1639	. . . 4 |- ( <. x , y >. e. A -> E. x <. x , y >. e. A )
4		opelxp 4784	. . . . 5 |- ( <. x , y >. e. ( dom A X. ran A ) <-> ( x e. dom A /\ y e. ran A ) )
5		vex 2818	. . . . . . 7 |- x e. _V
6	5	eldm2 4959	. . . . . 6 |- ( x e. dom A <-> E. y <. x , y >. e. A )
7		vex 2818	. . . . . . 7 |- y e. _V
8	7	elrn2 5004	. . . . . 6 |- ( y e. ran A <-> E. x <. x , y >. e. A )
9	6, 8	anbi12i 460	. . . . 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 184	. . . 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 417	. . 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 4847	1 |- ( Rel A -> A C_ ( dom A X. ran A ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1541   e. wcel 2205   C_ wss 3214   <.cop 3697   X. cxp 4752   dom cdm 4754   ran crn 4755   Rel wrel 4759

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  cnvssrndm  5289  cossxp  5290  relrelss  5294  relfld  5296  cnvexg  5305  fssxp  5535  oprabss  6147  resfunexgALT  6310  cofunexg  6311  fnexALT  6313  funexw  6314  erssxp  6803  znleval  14927