Theorem relssdmrn 5303

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 1643	. . . 4 |- ( <. x , y >. e. A -> E. y <. x , y >. e. A )
3		19.8a 1643	. . . 4 |- ( <. x , y >. e. A -> E. x <. x , y >. e. A )
4		opelxp 4799	. . . . 5 |- ( <. x , y >. e. ( dom A X. ran A ) <-> ( x e. dom A /\ y e. ran A ) )
5		vex 2824	. . . . . . 7 |- x e. _V
6	5	eldm2 4974	. . . . . 6 |- ( x e. dom A <-> E. y <. x , y >. e. A )
7		vex 2824	. . . . . . 7 |- y e. _V
8	7	elrn2 5019	. . . . . 6 |- ( y e. ran A <-> E. x <. x , y >. e. A )
9	6, 8	anbi12i 464	. . . . 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 421	. . 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 4862	1 |- ( Rel A -> A C_ ( dom A X. ran A ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   C_ wss 3220   <.cop 3708   X. cxp 4767   dom cdm 4769   ran crn 4770   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  cnvssrndm  5304  cossxp  5305  relrelss  5309  relfld  5311  cnvexg  5320  fssxp  5550  oprabss  6164  resfunexgALT  6327  cofunexg  6328  fnexALT  6330  funexw  6331  erssxp  6820  znleval  14960