Theorem relssdmrn 4841

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 1482	. . . 4 |- ( <. x , y >. e. A -> E. y <. x , y >. e. A )
3		19.8a 1482	. . . 4 |- ( <. x , y >. e. A -> E. x <. x , y >. e. A )
4		opelxp 4374	. . . . 5 |- ( <. x , y >. e. ( dom A X. ran A ) <-> ( x e. dom A /\ y e. ran A ) )
5		vex 2560	. . . . . . 7 |- x e. _V
6	5	eldm2 4533	. . . . . 6 |- ( x e. dom A <-> E. y <. x , y >. e. A )
7		vex 2560	. . . . . . 7 |- y e. _V
8	7	elrn2 4576	. . . . . 6 |- ( y e. ran A <-> E. x <. x , y >. e. A )
9	6, 8	anbi12i 433	. . . . 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 173	. . . 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 394	. . 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 4432	1 |- ( Rel A -> A C_ ( dom A X. ran A ) )

Syntax hints:   -> wi 4   /\ wa 97   E.wex 1381   e. wcel 1393   C_ wss 2917   <.cop 3378   X. cxp 4343   dom cdm 4345   ran crn 4346   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  cnvssrndm  4842  cossxp  4843  relrelss  4844  relfld  4846  cnvexg  4855  fssxp  5058  oprabss  5590  resfunexgALT  5737  cofunexg  5738  fnexALT  5740  erssxp  6129