Theorem elreldm 4503

Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)

Assertion

Ref	Expression
elreldm	|- (( Rel A /\ B e. A) -> |^| |^|B e. dom A)

Proof of Theorem elreldm

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

Step	Hyp	Ref	Expression
1		df-rel 4295	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V))
2		ssel 2933	. . . . 5 |- (A C_ ( _V X. _V) -> (B e. A -> B e. ( _V X. _V)))
3	1, 2	sylbi 114	. . . 4 |- ( Rel A -> (B e. A -> B e. ( _V X. _V)))
4		elvv 4345	. . . 4 |- (B e. ( _V X. _V) <-> E.xE.y B = <.x , y >.)
5	3, 4	syl6ib 150	. . 3 |- ( Rel A -> (B e. A -> E.xE.y B = <.x , y >.))
6		eleq1 2097	. . . . . 6 |- (B = <.x , y >. -> (B e. A <-> <.x , y >. e. A))
7		vex 2554	. . . . . . 7 |- x e. _V
8		vex 2554	. . . . . . 7 |- y e. _V
9	7, 8	opeldm 4481	. . . . . 6 |- ( <.x , y >. e. A -> x e. dom A)
10	6, 9	syl6bi 152	. . . . 5 |- (B = <.x , y >. -> (B e. A -> x e. dom A))
11		inteq 3609	. . . . . . . 8 |- (B = <.x , y >. -> |^|B = |^| <.x , y >.)
12	11	inteqd 3611	. . . . . . 7 |- (B = <.x , y >. -> |^| |^|B = |^| |^| <.x , y >.)
13	7, 8	op1stb 4175	. . . . . . 7 |- |^| |^| <.x , y >. = x
14	12, 13	syl6eq 2085	. . . . . 6 |- (B = <.x , y >. -> |^| |^|B = x)
15	14	eleq1d 2103	. . . . 5 |- (B = <.x , y >. -> ( |^| |^|B e. dom A <-> x e. dom A))
16	10, 15	sylibrd 158	. . . 4 |- (B = <.x , y >. -> (B e. A -> |^| |^|B e. dom A))
17	16	exlimivv 1773	. . 3 |- (E.xE.y B = <.x , y >. -> (B e. A -> |^| |^|B e. dom A))
18	5, 17	syli 33	. 2 |- ( Rel A -> (B e. A -> |^| |^|B e. dom A))
19	18	imp 115	1 |- (( Rel A /\ B e. A) -> |^| |^|B e. dom A)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551   C_ wss 2911   <.cop 3370   |^|cint 3606   X. cxp 4286   dom cdm 4288   Rel wrel 4293

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298

This theorem is referenced by:  1stdm  5750  fundmen  6222