Theorem relssdmrn 5190

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 1604	. . . 4 |- ( <. x , y >. e. A -> E. y <. x , y >. e. A )
3		19.8a 1604	. . . 4 |- ( <. x , y >. e. A -> E. x <. x , y >. e. A )
4		opelxp 4693	. . . . 5 |- ( <. x , y >. e. ( dom A X. ran A ) <-> ( x e. dom A /\ y e. ran A ) )
5		vex 2766	. . . . . . 7 |- x e. _V
6	5	eldm2 4864	. . . . . 6 |- ( x e. dom A <-> E. y <. x , y >. e. A )
7		vex 2766	. . . . . . 7 |- y e. _V
8	7	elrn2 4908	. . . . . 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 4755	1 |- ( Rel A -> A C_ ( dom A X. ran A ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1506   e. wcel 2167   C_ wss 3157   <.cop 3625   X. cxp 4661   dom cdm 4663   ran crn 4664   Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  cnvssrndm  5191  cossxp  5192  relrelss  5196  relfld  5198  cnvexg  5207  fssxp  5425  oprabss  6008  resfunexgALT  6165  cofunexg  6166  fnexALT  6168  funexw  6169  erssxp  6615  znleval  14209