Theorem elreldm 4853

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 4633	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
2		ssel 3149	. . . . 5 |- ( A C_ ( _V X. _V ) -> ( B e. A -> B e. ( _V X. _V ) ) )
3	1, 2	sylbi 121	. . . 4 |- ( Rel A -> ( B e. A -> B e. ( _V X. _V ) ) )
4		elvv 4688	. . . 4 |- ( B e. ( _V X. _V ) <-> E. x E. y B = <. x , y >. )
5	3, 4	imbitrdi 161	. . 3 |- ( Rel A -> ( B e. A -> E. x E. y B = <. x , y >. ) )
6		eleq1 2240	. . . . . 6 |- ( B = <. x , y >. -> ( B e. A <-> <. x , y >. e. A ) )
7		vex 2740	. . . . . . 7 |- x e. _V
8		vex 2740	. . . . . . 7 |- y e. _V
9	7, 8	opeldm 4830	. . . . . 6 |- ( <. x , y >. e. A -> x e. dom A )
10	6, 9	syl6bi 163	. . . . 5 |- ( B = <. x , y >. -> ( B e. A -> x e. dom A ) )
11		inteq 3847	. . . . . . . 8 |- ( B = <. x , y >. -> |^| B = |^| <. x , y >. )
12	11	inteqd 3849	. . . . . . 7 |- ( B = <. x , y >. -> |^| |^| B = |^| |^| <. x , y >. )
13	7, 8	op1stb 4478	. . . . . . 7 |- |^| |^| <. x , y >. = x
14	12, 13	eqtrdi 2226	. . . . . 6 |- ( B = <. x , y >. -> |^| |^| B = x )
15	14	eleq1d 2246	. . . . 5 |- ( B = <. x , y >. -> ( |^| |^| B e. dom A <-> x e. dom A ) )
16	10, 15	sylibrd 169	. . . 4 |- ( B = <. x , y >. -> ( B e. A -> |^| |^| B e. dom A ) )
17	16	exlimivv 1896	. . 3 |- ( E. x E. y B = <. x , y >. -> ( B e. A -> |^| |^| B e. dom A ) )
18	5, 17	syli 37	. 2 |- ( Rel A -> ( B e. A -> |^| |^| B e. dom A ) )
19	18	imp 124	1 |- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   C_ wss 3129   <.cop 3595   |^|cint 3844   X. cxp 4624   dom cdm 4626   Rel wrel 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-int 3845  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-dm 4636

This theorem is referenced by:  1stdm  6182  fundmen  6805